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

<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</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"> ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> implies  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c2">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">m</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>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">m</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</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> for <font color="Olive" title="b1">p</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <br/>  for <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <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="b1">p</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b3">N</font> </span> } </span>  holds <br/> ex <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b4">r2</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b3">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b4">r2</font> &amp; (  for <font color="Olive" title="b5">o</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Olive" title="b3">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b5">o</font></span>)</span> ) );<br/>
<span class="kw">assume </span><a NAME="E1:33"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 ; <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#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c2">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">m</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>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">m</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span></span><br/>

<a NAME="E2:33"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/>
<a NAME="E3:33"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<a NAME="E4:33"/>
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</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/>
<a NAME="E5:33"/><span class="kw">then </span>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E6:33"/><span class="kw">then </span>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:33"/><span class="kw">then </span>
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:33"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <font color="Maroon" title="c1">m</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="catalan2.html#T26" target="_self" title="CATALAN2:th.26">Th30</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">W</font> =  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">m</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="card_fin.html#T16" title="CARD_FIN:th.16">CARD_FIN:16</a></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="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">CH</font> =  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>);<br/>
<a NAME="E10:33"/><span class="lab"><font color="Green" title="E33">A4</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"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>)</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> <font color="Maroon" title="c3">W</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="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>)</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> <font color="Maroon" title="c3">W</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1"/><span class="lab"><font color="Green" title="E34">A5</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"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>)</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> <font color="Maroon" title="c3">W</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<a NAME="E2:33_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></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="E4:33_1"/><span class="lab"><font color="Green" title="E35">A6</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="E5:33_1"/><span class="lab"><font color="Green" title="E36">A7</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="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E6:33_1"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">and </span><br/><a NAME="E7:33_1"/><span class="lab"><font color="Green" title="E38">A9</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="c1">m</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="card_fin.html#T40" title="CARD_FIN:th.40">CARD_FIN:40</a></span>;<br/>
<a NAME="E8:33_1"/>
 not <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> (<font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E9:33_1"/><span class="kw">then </span>
 not <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="#E5:33_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E7:33_1"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="catalan2.html#D2" target="_self" title="CATALAN2:def.2">Def2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">o</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E11:33_1"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</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">p</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">N</font> </span> } </span>  &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Maroon" title="c8">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="catalan2.html#T15" target="_self" title="CATALAN2:th.15">Th19</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">r2</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="E13:33_1"/><span class="lab"><font color="Green" title="E40">A11</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c10">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="catalan2.html#T1" target="_self" title="CATALAN2:th.1">Th5</a></span>;<br/>
<a NAME="E14:33_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <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="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T70" title="RELAT_1:th.70">RELAT_1:70</a></span>;<br/>
<a NAME="E15:33_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">r1</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="E17:33_1"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <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>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">r1</font>
 <span class="kw">and </span><br/><a NAME="E18:33_1"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">r1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">and </span><br/><a NAME="E19:33_1"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> holds <br/><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"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">i</font></span>)</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> <span class="p2">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">i</font>
 <span class="kw">and </span><br/><a NAME="E20:33_1"/><span class="lab"><font color="Green" title="E44">A15</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> holds <br/><font color="Maroon" title="c11">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="catalan2.html#T25" target="_self" title="CATALAN2:th.25">Th29</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c12">R</font> = <font color="Maroon" title="c11">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c10">r2</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="c12">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c12">R</font>] )</span><br/>

<a NAME="E21:33_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">p</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> <font color="Maroon" title="c11">r1</font></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="c10">r2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33_1"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E13:33_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="afinsq_1.html#T17" title="AFINSQ_1:th.17">AFINSQ_1:17</a></span>;<br/>
<a NAME="E22:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1"><span class="lab"><font color="Green" title="E36">A7</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="E23:33_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">r2</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="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:33_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="afinsq_1.html#T25" title="AFINSQ_1:th.25">AFINSQ_1:25</a></span>;<br/>
<a NAME="E24:33_1"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">r2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E25:33_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">r1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:33_1"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E26:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A17</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T26" title="AFINSQ_1:th.26">AFINSQ_1:26</a></span>;<br/>
<a NAME="E27:33_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <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="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c11">r1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">r1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r1</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="E28:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A18</font></span>: 
<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">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</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="c11">r1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33_1"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E19:33_1"><span class="lab"><font color="Green" title="E43">A14</font></span></a></span>;<br/>
<a NAME="E29:33_1"/><span class="lab"><font color="Green" title="E48">A19</font></span>: 
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</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="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E30:33_1"/><span class="kw">then </span>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</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="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<a NAME="E31:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A20</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T74" title="RELAT_1:th.74">RELAT_1:74</a></span>;<br/>
<a NAME="E32:33_1"/><span class="lab"><font color="Green" title="E50">A21</font></span>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:33_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<a NAME="E33:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A22</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <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> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</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="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E34:33_1"/><span class="lab"><font color="Green" title="E52">A23</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <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> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</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="#E32:33_1"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E35:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A24</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T56" title="AFINSQ_1:th.56">AFINSQ_1:56</a></span>;<br/>
<a NAME="E36:33_1"/>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</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="#E29:33_1"><span class="lab"><font color="Green" title="E48">A19</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E37:33_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:33_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E38:33_1"/><span class="kw">then </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">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <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> <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">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span></span>)</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> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:33_1"><span class="lab"><font color="Green" title="E53">A24</font></span></a>, <a class="txt" href="#E31:33_1"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="ref" href="afinsq_2.html#T55" title="AFINSQ_2:th.55">AFINSQ_2:55</a></span>;<br/>
<a NAME="E39:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A25</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">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <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="c8">o</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="#E11:33_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="afinsq_2.html#T53" title="AFINSQ_2:th.53">AFINSQ_2:53</a></span>;<br/>
<a NAME="E40:33_1"/>
<font color="Maroon" title="c1">m</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_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</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="c10">r2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_1"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E13:33_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="afinsq_2.html#T55" title="AFINSQ_2:th.55">AFINSQ_2:55</a></span>;<br/>
<a NAME="E41:33_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c11">r1</font></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="c10">r2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:33_1"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E33:33_1"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="txt" href="#E39:33_1"><span class="lab"><font color="Green" title="E54">A25</font></span></a></span>;<br/>
<a NAME="E42:33_1"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 
<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/>
<span class="kw">hence </span><a NAME="E43:33_1"/>
<font color="Maroon" title="c12">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:33_1"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="txt" href="#E26:33_1"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="ref" href="card_fin.html#T40" title="CARD_FIN:th.40">CARD_FIN:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c12">R</font>]</span><br/>

<span class="kw">thus </span><a NAME="E44:33_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c12">R</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">p9</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p9</font> &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <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="Maroon" title="c13">p9</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">N</font> </span> } </span>  holds <br/> ex <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( <font color="Maroon" title="c12">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b3">r2</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b3">r2</font> &amp; (  for <font color="Olive" title="b4">o</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Olive" title="b2">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b4">o</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c14">k</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"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p9</font> &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</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="c13">p9</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">N</font> </span> } </span>  implies  ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( <font color="Maroon" title="c12">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b2">r2</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b2">r2</font> &amp; (  for <font color="Olive" title="b3">o</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Olive" title="b1">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">o</font></span>)</span> ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:33_1_1"/><span class="lab"><font color="Green" title="E55">A26</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p9</font> &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</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="c13">p9</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">N</font> </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">r1</font>, <font color="Olive" title="b2">r2</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( <font color="Maroon" title="c12">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b2">r2</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b2">r2</font> &amp; (  for <font color="Olive" title="b3">o</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Olive" title="b1">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b3">o</font></span>)</span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c15">q9</font> = <font color="Maroon" title="c13">p9</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">r1</font>
; <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">r2</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( <font color="Maroon" title="c12">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b1">r2</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b1">r2</font> &amp; (  for <font color="Olive" title="b2">o</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Maroon" title="c11">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">o</font></span>)</span> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c10">r2</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="c12">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c10">r2</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c10">r2</font> &amp; (  for <font color="Olive" title="b1">o</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Maroon" title="c11">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">o</font></span>)</span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:33_1_1"/>
( <font color="Maroon" title="c12">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c10">r2</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c10">r2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E11:33_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E17:33_1"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E13:33_1"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E32:33_1"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="txt" href="#E1:33_1_1"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">o</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Maroon" title="c11">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">o</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E3:33_1_1"/>
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Maroon" title="c11">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">i</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"> ( <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 implies <font color="Maroon" title="c11">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1_1_1"/>
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ; <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="c11">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span></span><br/>

<a NAME="E2:33_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A27</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E11:33_1"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E33:33_1"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="txt" href="#E1:33_1_1"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E3:33_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:33_1"><span class="lab"><font color="Green" title="E44">A15</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:33_1_1_1"/>
<font color="Maroon" title="c11">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p9</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c16">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E1:33_1_1"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="txt" href="#E2:33_1_1_1"><span class="lab"><font color="Green" title="E56">A27</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</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>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c6">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c2">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>)</span>)</span></span>)</span>,<font color="Maroon" title="c3">W</font><span class="kw"> such that </span><br/><a NAME="E12:33"/><span class="lab"><font color="Green" title="E34">A28</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"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>)</span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c6">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="#E10:33"><span class="lab"><font color="Green" title="E33">A4</font></span></a>);<br/></span>
<a NAME="E13:33"/>
<font color="Maroon" title="c3">W</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="c6">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W</font> or <font color="Maroon" title="c7">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="c6">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_2"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W</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">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="c6">F</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">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="E3:33_2"/><span class="lab"><font color="Green" title="E35">A29</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">and </span><br/><a NAME="E4:33_2"/><span class="lab"><font color="Green" title="E36">A30</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E5:33_2"/><span class="lab"><font color="Green" title="E37">A31</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">and </span><br/><a NAME="E6:33_2"/><span class="lab"><font color="Green" title="E38">A32</font></span>: 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_fin.html#T40" title="CARD_FIN:th.40">CARD_FIN:40</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">M</font> = <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="c8">p</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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">N</font> </span> } </span> ;<br/>
<a NAME="E7:33_2"/>
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</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/>
<a NAME="E8:33_2"/><span class="kw">then </span>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/>
<a NAME="E9:33_2"/><span class="kw">then </span>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:33_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="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_2"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="txt" href="#E6:33_2"><span class="lab"><font color="Green" title="E38">A32</font></span></a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a>, <a class="ref" href="relat_1.html#T69" title="RELAT_1:th.69">RELAT_1:69</a></span>;<br/>
<a NAME="E11:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A33</font></span>: 
<font color="Maroon" title="c2">n</font> <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="c8">p</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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">N</font> </span> } </span> 
 
;<br/>
<a NAME="E12:33_2"/>
<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="c8">p</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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">N</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">y</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="c10">y</font> <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="c8">p</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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">N</font> </span> } </span>  or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_2_1"/>
<font color="Maroon" title="c10">y</font> <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="c8">p</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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">N</font> </span> } </span> 
 ; <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="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<a NAME="E2:33_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">i</font> being    <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>  st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> &amp; 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="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:33_2_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">M</font> = <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="c8">p</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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">N</font> </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:33_2"><span class="lab"><font color="Green" title="E39">A33</font></span></a></span>;<br/>
<a NAME="E14:33_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font> &amp;  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">k</font></span>)</span> <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="Olive" title="b1">k</font> &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c11">mm</font> =  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font>;<br/>
<a NAME="E1:33_2_2"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></span>;<br/>
<a NAME="E2:33_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A34</font></span>: 
 ex <font color="Olive" title="b1">o</font> being    <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>  st <br/>(  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> &amp; 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="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">o</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">o</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">m1</font> = <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1 as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/>
<a NAME="E4:33_2_2"/><span class="lab"><font color="Green" title="E41">A35</font></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="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">m1</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="#E2:33_2_2"><span class="lab"><font color="Green" title="E40">A34</font></span></a></span>;<br/>
<a NAME="E5:33_2_2"/><span class="lab"><font color="Green" title="E42">A36</font></span>: 
<font color="Maroon" title="c12">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">m1</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/>
<a NAME="E6:33_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">m1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<a NAME="E7:33_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A37</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T74" title="RELAT_1:th.74">RELAT_1:74</a></span>;<br/>
<a NAME="E8:33_2_2"/>
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_2"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="txt" href="#E11:33_2"><span class="lab"><font color="Green" title="E39">A33</font></span></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></span>;<br/>
<a NAME="E9:33_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A38</font></span>: 
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<a NAME="E10:33_2_2"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E11:33_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_2_2"><span class="lab"><font color="Green" title="E42">A36</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E12:33_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A39</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E13:33_2_2"/>
<font color="Maroon" title="c12">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">m1</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/>
<a NAME="E14:33_2_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c12">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></span>;<br/>
<a NAME="E15:33_2_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="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">m1</font>
 
;<br/>
<a NAME="E16:33_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A40</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"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">m1</font></span>)</span></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> <font color="Maroon" title="c8">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">m1</font> &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">m1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">m1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">S</font> =  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font></span>)</span></span>)</span> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">S</font>
; <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">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c13">S</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font> &amp;  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c13">S</font></span>)</span> <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="c13">S</font> &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c13">S</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">p</font> )</span><br/>

<a NAME="E18:33_2_2"/><span class="lab"><font color="Green" title="E47">A41</font></span>: 
 <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_2"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="txt" href="#E11:33_2"><span class="lab"><font color="Green" title="E39">A33</font></span></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></span>;<br/>
<a NAME="E19:33_2_2"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m1</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="#E9:33_2_2"><span class="lab"><font color="Green" title="E44">A38</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E20:33_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">m1</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">m1</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T56" title="AFINSQ_1:th.56">AFINSQ_1:56</a></span>;<br/>
<a NAME="E21:33_2_2"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font></span>)</span></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">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c12">m1</font></span>)</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> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">m1</font></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33_2_2"><span class="lab"><font color="Green" title="E45">A39</font></span></a>, <a class="txt" href="#E7:33_2_2"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="ref" href="afinsq_2.html#T55" title="AFINSQ_2:th.55">AFINSQ_2:55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:33_2_2"/>
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c13">S</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font> &amp;  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c13">S</font></span>)</span> <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="c13">S</font> &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c13">S</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:33_2_2"><span class="lab"><font color="Green" title="E47">A41</font></span></a>, <a class="txt" href="#E16:33_2_2"><span class="lab"><font color="Green" title="E46">A40</font></span></a>, <a class="txt" href="#E4:33_2_2"><span class="lab"><font color="Green" title="E41">A35</font></span></a>, <a class="ref" href="nat_1.html#T9" title="NAT_1:th.9">NAT_1:9</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">then </span><span class="kw">consider </span><font color="Maroon" title="c11">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E16:33_2"/><span class="lab"><font color="Green" title="E40">A42</font></span>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</a> <font color="Maroon" title="c10">M</font>
 <span class="kw">and </span><br/><a NAME="E17:33_2"/><span class="lab"><font color="Green" title="E41">A43</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="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <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="c11">k</font>
 <span class="kw">and </span><br/><a NAME="E18:33_2"/><span class="lab"><font color="Green" title="E42">A44</font></span>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">p</font>
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">k1</font> = <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">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="E20:33_2"/><span class="lab"><font color="Green" title="E43">A45</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="catalan2.html#T1" target="_self" title="CATALAN2:th.1">Th5</a></span>;<br/>
<a NAME="E21:33_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <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="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T70" title="RELAT_1:th.70">RELAT_1:70</a></span>;<br/>
<a NAME="E22:33_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_2"><span class="lab"><font color="Green" title="E37">A31</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">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="E24:33_2"/><span class="lab"><font color="Green" title="E44">A46</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c14">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E25:33_2"/><span class="lab"><font color="Green" title="E45">A47</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">and </span><br/><a NAME="E26:33_2"/><span class="lab"><font color="Green" title="E46">A48</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> holds <br/><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"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">i</font></span>)</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> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">i</font>
 <span class="kw">and </span><br/><a NAME="E27:33_2"/><span class="lab"><font color="Green" title="E47">A49</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> holds <br/><font color="Maroon" title="c14">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="catalan2.html#T25" target="_self" title="CATALAN2:th.25">Th29</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">rq</font> = <font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font>;<br/>
<a NAME="E28:33_2"/><span class="lab"><font color="Green" title="E48">A50</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</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">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</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="c13">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:33_2"><span class="lab"><font color="Green" title="E44">A46</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="E29:33_2"/><span class="lab"><font color="Green" title="E49">A51</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c13">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_2"><span class="lab"><font color="Green" title="E38">A32</font></span></a>, <a class="txt" href="#E17:33_2"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="txt" href="#E20:33_2"><span class="lab"><font color="Green" title="E43">A45</font></span></a>, <a class="ref" href="afinsq_2.html#T55" title="AFINSQ_2:th.55">AFINSQ_2:55</a></span>;<br/>
<a NAME="E30:33_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</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">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</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="c13">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:33_2"><span class="lab"><font color="Green" title="E44">A46</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="E31:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A52</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <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="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:33_2"><span class="lab"><font color="Green" title="E43">A45</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="E32:33_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <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="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c14">r</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r</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="E33:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A53</font></span>: 
<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="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</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="c14">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:33_2"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="txt" href="#E26:33_2"><span class="lab"><font color="Green" title="E46">A48</font></span></a></span>;<br/>
<a NAME="E34:33_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">q</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="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:33_2"><span class="lab"><font color="Green" title="E43">A45</font></span></a>, <a class="ref" href="afinsq_1.html#T25" title="AFINSQ_1:th.25">AFINSQ_1:25</a></span>;<br/>
<a NAME="E35:33_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_2"><span class="lab"><font color="Green" title="E37">A31</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E36:33_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:33_2"><span class="lab"><font color="Green" title="E45">A47</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E37:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A54</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T26" title="AFINSQ_1:th.26">AFINSQ_1:26</a></span>;<br/>
<a NAME="E38:33_2"/><span class="lab"><font color="Green" title="E53">A55</font></span>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:33_2"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<a NAME="E39:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A56</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <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> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</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="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E40:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A57</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <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="c14">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:33_2"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="ref" href="afinsq_1.html#T57" title="AFINSQ_1:th.57">AFINSQ_1:57</a></span>;<br/>
<a NAME="E41:33_2"/><span class="lab"><font color="Green" title="E56">A58</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</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="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E42:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A59</font></span>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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> <font color="Maroon" title="c14">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33_2"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="txt" href="#E33:33_2"><span class="lab"><font color="Green" title="E51">A53</font></span></a>, <a class="txt" href="#E39:33_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a></span>;<br/>
<a NAME="E43:33_2"/><span class="lab"><font color="Green" title="E58">A60</font></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> <font color="Maroon" title="c14">r</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</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="#E17:33_2"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="txt" href="#E33:33_2"><span class="lab"><font color="Green" title="E51">A53</font></span></a>, <a class="txt" href="#E39:33_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a>, <a class="txt" href="#E41:33_2"><span class="lab"><font color="Green" title="E56">A58</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E45:33_2"/><span class="lab"><font color="Green" title="E59">A61</font></span>: 
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</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"><span class="p3">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">N</font> </span> } </span>  &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> &amp; <font color="Maroon" title="c16">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:33_2"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="txt" href="#E40:33_2"><span class="lab"><font color="Green" title="E55">A57</font></span></a>, <a class="ref" href="catalan2.html#T2" target="_self" title="CATALAN2:th.2">Th6</a>, <a class="ref" href="catalan2.html#T15" target="_self" title="CATALAN2:th.15">Th19</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">j1</font> = <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1;<br/>
<a NAME="E46:33_2"/><span class="lab"><font color="Green" title="E60">A62</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</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">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</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="c13">q</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="E47:33_2"/>
 not <font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">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="#E43:33_2"><span class="lab"><font color="Green" title="E58">A60</font></span></a>, <a class="txt" href="#E40:33_2"><span class="lab"><font color="Green" title="E55">A57</font></span></a>, <a class="ref" href="catalan2.html#T2" target="_self" title="CATALAN2:th.2">Th6</a></span>;<br/>
<a NAME="E48:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A63</font></span>: 
 not <font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">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="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="catalan2.html#T20" target="_self" title="CATALAN2:th.20">Th24</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">rqj</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>;<br/>
<a NAME="E49:33_2"/>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</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> <font color="Maroon" title="c14">r</font></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="c13">q</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="E50:33_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_2"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="txt" href="#E17:33_2"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="txt" href="#E33:33_2"><span class="lab"><font color="Green" title="E51">A53</font></span></a>, <a class="txt" href="#E39:33_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a>, <a class="txt" href="#E31:33_2"><span class="lab"><font color="Green" title="E50">A52</font></span></a>, <a class="txt" href="#E37:33_2"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="txt" href="#E29:33_2"><span class="lab"><font color="Green" title="E49">A51</font></span></a>, <a class="ref" href="card_fin.html#T40" title="CARD_FIN:th.40">CARD_FIN:40</a></span>;<br/>
<a NAME="E51:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A64</font></span>: 
<font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:33_2"><span class="lab"><font color="Green" title="E61">A63</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">r1</font>, <font color="Maroon" title="c20">r2</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="E53:33_2"/><span class="lab"><font color="Green" title="E63">A65</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font>
 <span class="kw">and </span><br/><a NAME="E54:33_2"/><span class="lab"><font color="Green" title="E64">A66</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">and </span><br/><a NAME="E55:33_2"/><span class="lab"><font color="Green" title="E65">A67</font></span>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font> )
 <span class="kw">and </span><br/><a NAME="E56:33_2"/><span class="lab"><font color="Green" title="E66">A68</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Maroon" title="c19">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E45:33_2"><span class="lab"><font color="Green" title="E59">A61</font></span></a></span>;<br/>
<a NAME="E57:33_2"/><span class="lab"><font color="Green" title="E67">A69</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</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> <font color="Maroon" title="c19">r1</font></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="c20">r2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:33_2"><span class="lab"><font color="Green" title="E65">A67</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="E58:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A70</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:33_2"><span class="lab"><font color="Green" title="E48">A50</font></span></a>, <a class="txt" href="#E46:33_2"><span class="lab"><font color="Green" title="E60">A62</font></span></a>, <a class="ref" href="afinsq_1.html#T17" title="AFINSQ_1:th.17">AFINSQ_1:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">K</font> = <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"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">N</font> </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:33_2"><span class="lab"><font color="Green" title="E59">A61</font></span></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></span>;<br/>
<a NAME="E60:33_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <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="c14">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:33_2"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="txt" href="#E39:33_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a>, <a class="ref" href="afinsq_1.html#T57" title="AFINSQ_1:th.57">AFINSQ_1:57</a></span>;<br/>
<a NAME="E61:33_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:33_2"><span class="lab"><font color="Green" title="E57">A59</font></span></a></span>;<br/>
<a NAME="E62:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A71</font></span>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</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="#E45:33_2"><span class="lab"><font color="Green" title="E59">A61</font></span></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></span>;<br/>
<a NAME="E63:33_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</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="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/>
<a NAME="E64:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A72</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <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="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T74" title="RELAT_1:th.74">RELAT_1:74</a></span>;<br/>
<a NAME="E65:33_2"/>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:33_2"><span class="lab"><font color="Green" title="E59">A61</font></span></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></span>;<br/>
<a NAME="E66:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A73</font></span>: 
 ex <font color="Olive" title="b1">N</font> being    <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>  st <br/>( <font color="Olive" title="b1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; 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"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">N</font> )
 
;<br/>
<a NAME="E67:33_2"/>
<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"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</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> <span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:33_2"><span class="lab"><font color="Green" title="E46">A48</font></span></a>, <a class="txt" href="#E39:33_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a>, <a class="txt" href="#E62:33_2"><span class="lab"><font color="Green" title="E69">A71</font></span></a></span>;<br/>
<a NAME="E68:33_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="c14">r</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <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> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E64:33_2"><span class="lab"><font color="Green" title="E70">A72</font></span></a></span>;<br/>
<a NAME="E69:33_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:33_2"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="txt" href="#E39:33_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a>, <a class="txt" href="#E62:33_2"><span class="lab"><font color="Green" title="E69">A71</font></span></a>, <a class="txt" href="#E66:33_2"><span class="lab"><font color="Green" title="E71">A73</font></span></a>, <a class="ref" href="afinsq_1.html#T58" title="AFINSQ_1:th.58">AFINSQ_1:58</a></span>;<br/>
<a NAME="E70:33_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E71:33_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M</font>
 
;<br/>
<a NAME="E72:33_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</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="#E16:33_2"><span class="lab"><font color="Green" title="E40">A42</font></span></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></span>;<br/>
<a NAME="E73:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A74</font></span>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">j</font></span>)</span> <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">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</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="#E62:33_2"><span class="lab"><font color="Green" title="E69">A71</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E74:33_2"/><span class="lab"><font color="Green" title="E73">A75</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:33_2"><span class="lab"><font color="Green" title="E48">A50</font></span></a>, <a class="ref" href="afinsq_1.html#D3" title="AFINSQ_1:def.3">AFINSQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">i</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"> ( <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:33_2_3"/><span class="lab"><font color="Green" title="E74">A76</font></span>: 
<font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font></span>)</span>
 ; <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="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_2_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:33_2_3_1_1"/>
( <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font> or <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_2_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:33_2_3_1_1_1"/><span class="lab"><font color="Green" title="E75">A77</font></span>: 
<font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font>
 ; <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="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span><br/><div class="add"><a NAME="E2:33_2_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A78</font></span>: 
( <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">r1</font> &amp; <font color="Maroon" title="c19">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:33_2"><span class="lab"><font color="Green" title="E64">A66</font></span></a>, <a class="txt" href="#E56:33_2"><span class="lab"><font color="Green" title="E66">A68</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/><a NAME="E3:33_2_3_1_1_1"/><span class="lab"><font color="Green" title="E77">A79</font></span>: 
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_2_3_1_1_1"><span class="lab"><font color="Green" title="E75">A77</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/><a NAME="E4:33_2_3_1_1_1"/><span class="lab"><font color="Green" title="E78">A80</font></span>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:33_2"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="txt" href="#E54:33_2"><span class="lab"><font color="Green" title="E64">A66</font></span></a>, <a class="txt" href="#E73:33_2"><span class="lab"><font color="Green" title="E72">A74</font></span></a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/><a NAME="E5:33_2_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A81</font></span>: 
<font color="Maroon" title="c14">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:33_2"><span class="lab"><font color="Green" title="E47">A49</font></span></a>, <a class="txt" href="#E3:33_2_3_1_1_1"><span class="lab"><font color="Green" title="E77">A79</font></span></a></span>;<br/><a NAME="E6:33_2_3_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:33_2"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="txt" href="#E3:33_2_3_1_1_1"><span class="lab"><font color="Green" title="E77">A79</font></span></a>, <a class="txt" href="#E4:33_2_3_1_1_1"><span class="lab"><font color="Green" title="E78">A80</font></span></a>, <a class="ref" href="afinsq_1.html#D3" title="AFINSQ_1:def.3">AFINSQ_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E7:33_2_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_2_3_1_1_1"><span class="lab"><font color="Green" title="E79">A81</font></span></a>, <a class="txt" href="#E2:33_2_3_1_1_1"><span class="lab"><font color="Green" title="E76">A78</font></span></a>, <a class="ref" href="afinsq_1.html#D3" title="AFINSQ_1:def.3">AFINSQ_1:def 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_2_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:33_2_3_1_1_2"/><span class="lab"><font color="Green" title="E75">A82</font></span>: 
<font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font>
 ; <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="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span><br/><div class="add"><a NAME="E2:33_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A83</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c14">r</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:33_2"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="txt" href="#E39:33_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a>, <a class="txt" href="#E54:33_2"><span class="lab"><font color="Green" title="E64">A66</font></span></a>, <a class="txt" href="#E73:33_2"><span class="lab"><font color="Green" title="E72">A74</font></span></a>, <a class="txt" href="#E58:33_2"><span class="lab"><font color="Green" title="E68">A70</font></span></a>, <a class="txt" href="#E1:33_2_3"><span class="lab"><font color="Green" title="E74">A76</font></span></a>, <a class="ref" href="afinsq_1.html#T19" title="AFINSQ_1:th.19">AFINSQ_1:19</a></span>;<br/><a NAME="E3:33_2_3_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">r2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c14">r</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:33_2"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="txt" href="#E39:33_2"><span class="lab"><font color="Green" title="E54">A56</font></span></a>, <a class="txt" href="#E54:33_2"><span class="lab"><font color="Green" title="E64">A66</font></span></a>, <a class="txt" href="#E73:33_2"><span class="lab"><font color="Green" title="E72">A74</font></span></a>, <a class="txt" href="#E58:33_2"><span class="lab"><font color="Green" title="E68">A70</font></span></a>, <a class="txt" href="#E74:33_2"><span class="lab"><font color="Green" title="E73">A75</font></span></a>, <a class="txt" href="#E1:33_2_3"><span class="lab"><font color="Green" title="E74">A76</font></span></a>, <a class="txt" href="#E1:33_2_3_1_1_2"><span class="lab"><font color="Green" title="E75">A82</font></span></a>, <a class="ref" href="afinsq_1.html#T19" title="AFINSQ_1:th.19">AFINSQ_1:19</a></span>;<br/><span class="kw">hence </span><a NAME="E4:33_2_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:33_2"><span class="lab"><font color="Green" title="E65">A67</font></span></a>, <a class="txt" href="#E58:33_2"><span class="lab"><font color="Green" title="E68">A70</font></span></a>, <a class="txt" href="#E74:33_2"><span class="lab"><font color="Green" title="E73">A75</font></span></a>, <a class="txt" href="#E1:33_2_3"><span class="lab"><font color="Green" title="E74">A76</font></span></a>, <a class="txt" href="#E1:33_2_3_1_1_2"><span class="lab"><font color="Green" title="E75">A82</font></span></a>, <a class="txt" href="#E2:33_2_3_1_1_2"><span class="lab"><font color="Green" title="E76">A83</font></span></a>, <a class="ref" href="afinsq_1.html#T19" title="AFINSQ_1:th.19">AFINSQ_1:19</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:33_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
 ; <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="E76:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A84</font></span>: 
<font color="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:33_2"><span class="lab"><font color="Green" title="E67">A69</font></span></a>, <a class="txt" href="#E28:33_2"><span class="lab"><font color="Green" title="E48">A50</font></span></a>, <a class="txt" href="#E46:33_2"><span class="lab"><font color="Green" title="E60">A62</font></span></a>, <a class="ref" href="afinsq_1.html#T9" title="AFINSQ_1:th.9">AFINSQ_1:9</a>, <a class="ref" href="afinsq_1.html#T17" title="AFINSQ_1:th.17">AFINSQ_1:17</a></span>;<br/>
<a NAME="E77:33_2"/>
<font color="Maroon" title="c14">r</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:33_2"><span class="lab"><font color="Green" title="E62">A64</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E78:33_2"/>
<font color="Maroon" title="c7">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="c6">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_2"><span class="lab"><font color="Green" title="E35">A29</font></span></a>, <a class="txt" href="#E20:33_2"><span class="lab"><font color="Green" title="E43">A45</font></span></a>, <a class="txt" href="#E53:33_2"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="txt" href="#E76:33_2"><span class="lab"><font color="Green" title="E74">A84</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="E14:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A85</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E15:33"/><span class="lab"><font color="Green" title="E36">A86</font></span>: 
<font color="Maroon" title="c6">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c6">F</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c6">F</font> or  not <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">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"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c6">F</font> or  not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c6">F</font> or  not <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:33_3"/><span class="lab"><font color="Green" title="E37">A87</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">and </span><br/><a NAME="E2:33_3"/><span class="lab"><font color="Green" title="E38">A88</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">and </span><br/><a NAME="E3:33_3"/><span class="lab"><font color="Green" title="E39">A89</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font></span><br/>

<a NAME="E4:33_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_3"><span class="lab"><font color="Green" title="E37">A87</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">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="E6:33_3"/><span class="lab"><font color="Green" title="E40">A90</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">and </span><br/><a NAME="E7:33_3"/><span class="lab"><font color="Green" title="E41">A91</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E8:33_3"/><span class="lab"><font color="Green" title="E42">A92</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">and </span><br/><a NAME="E9:33_3"/><span class="lab"><font color="Green" title="E43">A93</font></span>: 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</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="card_fin.html#T40" title="CARD_FIN:th.40">CARD_FIN:40</a></span>;<br/>
<a NAME="E10:33_3"/>
 not <font color="Maroon" title="c9">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> (<font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_3"><span class="lab"><font color="Green" title="E37">A87</font></span></a>, <a class="txt" href="#E6:33_3"><span class="lab"><font color="Green" title="E40">A90</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E11:33_3"/><span class="kw">then </span>
 not <font color="Maroon" title="c9">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="#E7:33_3"><span class="lab"><font color="Green" title="E41">A91</font></span></a>, <a class="txt" href="#E9:33_3"><span class="lab"><font color="Green" title="E43">A93</font></span></a>, <a class="ref" href="catalan2.html#D2" target="_self" title="CATALAN2:def.2">Def2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">k1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E13:33_3"/><span class="lab"><font color="Green" title="E44">A94</font></span>: 
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</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="c9">p</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">N</font> </span> } </span>  &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c10">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_3"><span class="lab"><font color="Green" title="E42">A92</font></span></a>, <a class="ref" href="catalan2.html#T15" target="_self" title="CATALAN2:th.15">Th19</a></span>;<br/>
<a NAME="E14:33_3"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_3"><span class="lab"><font color="Green" title="E38">A88</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">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="E16:33_3"/><span class="lab"><font color="Green" title="E45">A95</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">and </span><br/><a NAME="E17:33_3"/><span class="lab"><font color="Green" title="E46">A96</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E18:33_3"/><span class="lab"><font color="Green" title="E47">A97</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">and </span><br/><a NAME="E19:33_3"/><span class="lab"><font color="Green" title="E48">A98</font></span>: 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</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="card_fin.html#T40" title="CARD_FIN:th.40">CARD_FIN:40</a></span>;<br/>
<a NAME="E20:33_3"/>
 not <font color="Maroon" title="c11">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="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_3"><span class="lab"><font color="Green" title="E38">A88</font></span></a>, <a class="txt" href="#E16:33_3"><span class="lab"><font color="Green" title="E45">A95</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E21:33_3"/><span class="kw">then </span>
 not <font color="Maroon" title="c11">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="#E17:33_3"><span class="lab"><font color="Green" title="E46">A96</font></span></a>, <a class="txt" href="#E19:33_3"><span class="lab"><font color="Green" title="E48">A98</font></span></a>, <a class="ref" href="catalan2.html#D2" target="_self" title="CATALAN2:def.2">Def2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">k2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E23:33_3"/><span class="lab"><font color="Green" title="E49">A99</font></span>: 
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K5" title="NAT_1:func.5">min*</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="c11">q</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">N</font> </span> } </span>  &amp; <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">q</font> &amp; <font color="Maroon" title="c12">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:33_3"><span class="lab"><font color="Green" title="E47">A97</font></span></a>, <a class="ref" href="catalan2.html#T15" target="_self" title="CATALAN2:th.15">Th19</a></span>;<br/>
<a NAME="E24:33_3"/><span class="lab"><font color="Green" title="E50">A100</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33_3"><span class="lab"><font color="Green" title="E46">A96</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">M</font> = <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="c11">q</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">N</font> </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:33_3"><span class="lab"><font color="Green" title="E49">A99</font></span></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">qk</font> = <font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">s1</font>, <font color="Maroon" title="c16">s2</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="E27:33_3"/><span class="lab"><font color="Green" title="E51">A101</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c16">s2</font>
 <span class="kw">and </span><br/><a NAME="E28:33_3"/><span class="lab"><font color="Green" title="E52">A102</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">and </span><br/><a NAME="E29:33_3"/><span class="lab"><font color="Green" title="E53">A103</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E30:33_3"/><span class="lab"><font color="Green" title="E54">A104</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c16">s2</font>
 <span class="kw">and </span><br/><a NAME="E31:33_3"/><span class="lab"><font color="Green" title="E55">A105</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Maroon" title="c15">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E2:33_3"><span class="lab"><font color="Green" title="E38">A88</font></span></a>, <a class="txt" href="#E16:33_3"><span class="lab"><font color="Green" title="E45">A95</font></span></a>, <a class="txt" href="#E23:33_3"><span class="lab"><font color="Green" title="E49">A99</font></span></a></span>;<br/>
<a NAME="E32:33_3"/><span class="lab"><font color="Green" title="E56">A106</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">q</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">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</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="c16">s2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:33_3"><span class="lab"><font color="Green" title="E54">A104</font></span></a>, <a class="ref" href="afinsq_1.html#T17" title="AFINSQ_1:th.17">AFINSQ_1:17</a></span>;<br/>
<a NAME="E33:33_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A107</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">s1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c16">s2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:33_3"><span class="lab"><font color="Green" title="E53">A103</font></span></a>, <a class="ref" href="afinsq_1.html#T17" title="AFINSQ_1:th.17">AFINSQ_1:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">K</font> = <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="c9">p</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">N</font> </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:33_3"><span class="lab"><font color="Green" title="E44">A94</font></span></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">pk</font> = <font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">r1</font>, <font color="Maroon" title="c20">r2</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="E36:33_3"/><span class="lab"><font color="Green" title="E58">A108</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font>
 <span class="kw">and </span><br/><a NAME="E37:33_3"/><span class="lab"><font color="Green" title="E59">A109</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">and </span><br/><a NAME="E38:33_3"/><span class="lab"><font color="Green" title="E60">A110</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E39:33_3"/><span class="lab"><font color="Green" title="E61">A111</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font>
 <span class="kw">and </span><br/><a NAME="E40:33_3"/><span class="lab"><font color="Green" title="E62">A112</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Maroon" title="c19">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E1:33_3"><span class="lab"><font color="Green" title="E37">A87</font></span></a>, <a class="txt" href="#E6:33_3"><span class="lab"><font color="Green" title="E40">A90</font></span></a>, <a class="txt" href="#E13:33_3"><span class="lab"><font color="Green" title="E44">A94</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E41:33_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font>
 ; <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="c21">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E43:33_3"/><span class="lab"><font color="Green" title="E63">A113</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">p</font>
 <span class="kw">and </span><br/><a NAME="E44:33_3"/><span class="lab"><font color="Green" title="E64">A114</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_3"><span class="lab"><font color="Green" title="E40">A90</font></span></a>, <a class="txt" href="#E7:33_3"><span class="lab"><font color="Green" title="E41">A91</font></span></a>, <a class="txt" href="#E16:33_3"><span class="lab"><font color="Green" title="E45">A95</font></span></a>, <a class="txt" href="#E17:33_3"><span class="lab"><font color="Green" title="E46">A96</font></span></a>, <a class="txt" href="#E32:33_3"><span class="lab"><font color="Green" title="E56">A106</font></span></a>, <a class="ref" href="afinsq_1.html#T9" title="AFINSQ_1:th.9">AFINSQ_1:9</a></span>;<br/>
<a NAME="E45:33_3"/><span class="lab"><font color="Green" title="E65">A115</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">p</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">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</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="c20">r2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:33_3"><span class="lab"><font color="Green" title="E61">A111</font></span></a>, <a class="ref" href="afinsq_1.html#T17" title="AFINSQ_1:th.17">AFINSQ_1:17</a></span>;<br/>
<a NAME="E46:33_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A116</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:33_3"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="ref" href="afinsq_1.html#T17" title="AFINSQ_1:th.17">AFINSQ_1:17</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:33_3_1_1"/>
( <font color="Maroon" title="c10">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k2</font> or <font color="Maroon" title="c10">k1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">k2</font> or <font color="Maroon" title="c10">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:33_3_1_1_1"/><span class="lab"><font color="Green" title="E67">A117</font></span>: 
<font color="Maroon" title="c10">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_3_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:33_3_1_1_1_1_1"/>
( <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> or <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_3_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:33_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E68">A118</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:33_3_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/><a NAME="E3:33_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A119</font></span>: 
( <font color="Maroon" title="c19">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c15">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">s1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c16">s2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:33_3"><span class="lab"><font color="Green" title="E59">A109</font></span></a>, <a class="txt" href="#E38:33_3"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="txt" href="#E28:33_3"><span class="lab"><font color="Green" title="E52">A102</font></span></a>, <a class="txt" href="#E1:33_3_1_1_1"><span class="lab"><font color="Green" title="E67">A117</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="E4:33_3_1_1_1_1_1_1"/>
( <font color="Maroon" title="c19">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> &amp; <font color="Maroon" title="c15">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:33_3"><span class="lab"><font color="Green" title="E59">A109</font></span></a>, <a class="txt" href="#E38:33_3"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="txt" href="#E40:33_3"><span class="lab"><font color="Green" title="E62">A112</font></span></a>, <a class="txt" href="#E31:33_3"><span class="lab"><font color="Green" title="E55">A105</font></span></a>, <a class="txt" href="#E1:33_3_1_1_1"><span class="lab"><font color="Green" title="E67">A117</font></span></a>, <a class="txt" href="#E1:33_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A118</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:33_3_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_3"><span class="lab"><font color="Green" title="E39">A89</font></span></a>, <a class="txt" href="#E36:33_3"><span class="lab"><font color="Green" title="E58">A108</font></span></a>, <a class="txt" href="#E27:33_3"><span class="lab"><font color="Green" title="E51">A101</font></span></a>, <a class="txt" href="#E44:33_3"><span class="lab"><font color="Green" title="E64">A114</font></span></a>, <a class="txt" href="#E3:33_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A119</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_3_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:33_3_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E68">A120</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:33_3_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A121</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">s1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c16">s2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_3"><span class="lab"><font color="Green" title="E41">A91</font></span></a>, <a class="txt" href="#E37:33_3"><span class="lab"><font color="Green" title="E59">A109</font></span></a>, <a class="txt" href="#E38:33_3"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="txt" href="#E17:33_3"><span class="lab"><font color="Green" title="E46">A96</font></span></a>, <a class="txt" href="#E28:33_3"><span class="lab"><font color="Green" title="E52">A102</font></span></a>, <a class="txt" href="#E33:33_3"><span class="lab"><font color="Green" title="E57">A107</font></span></a>, <a class="txt" href="#E43:33_3"><span class="lab"><font color="Green" title="E63">A113</font></span></a>, <a class="txt" href="#E1:33_3_1_1_1"><span class="lab"><font color="Green" title="E67">A117</font></span></a>, <a class="ref" href="afinsq_1.html#T19" title="AFINSQ_1:th.19">AFINSQ_1:19</a></span>;<br/><a NAME="E3:33_3_1_1_1_1_1_2"/>
( <font color="Maroon" title="c9">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">r2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">s2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">i</font> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_3"><span class="lab"><font color="Green" title="E41">A91</font></span></a>, <a class="txt" href="#E37:33_3"><span class="lab"><font color="Green" title="E59">A109</font></span></a>, <a class="txt" href="#E38:33_3"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="txt" href="#E39:33_3"><span class="lab"><font color="Green" title="E61">A111</font></span></a>, <a class="txt" href="#E28:33_3"><span class="lab"><font color="Green" title="E52">A102</font></span></a>, <a class="txt" href="#E29:33_3"><span class="lab"><font color="Green" title="E53">A103</font></span></a>, <a class="txt" href="#E30:33_3"><span class="lab"><font color="Green" title="E54">A104</font></span></a>, <a class="txt" href="#E24:33_3"><span class="lab"><font color="Green" title="E50">A100</font></span></a>, <a class="txt" href="#E43:33_3"><span class="lab"><font color="Green" title="E63">A113</font></span></a>, <a class="txt" href="#E1:33_3_1_1_1"><span class="lab"><font color="Green" title="E67">A117</font></span></a>, <a class="txt" href="#E1:33_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E68">A120</font></span></a>, <a class="ref" href="afinsq_1.html#T19" title="AFINSQ_1:th.19">AFINSQ_1:19</a></span>;<br/><span class="kw">hence </span><a NAME="E4:33_3_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_3"><span class="lab"><font color="Green" title="E39">A89</font></span></a>, <a class="txt" href="#E36:33_3"><span class="lab"><font color="Green" title="E58">A108</font></span></a>, <a class="txt" href="#E38:33_3"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="txt" href="#E27:33_3"><span class="lab"><font color="Green" title="E51">A101</font></span></a>, <a class="txt" href="#E46:33_3"><span class="lab"><font color="Green" title="E66">A116</font></span></a>, <a class="txt" href="#E43:33_3"><span class="lab"><font color="Green" title="E63">A113</font></span></a>, <a class="txt" href="#E44:33_3"><span class="lab"><font color="Green" title="E64">A114</font></span></a>, <a class="txt" href="#E1:33_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E68">A120</font></span></a>, <a class="txt" href="#E2:33_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E69">A121</font></span></a>, <a class="ref" href="afinsq_1.html#T19" title="AFINSQ_1:th.19">AFINSQ_1:19</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:33_3_1_1_1"/>
contradiction
 ; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:33_3_1_1_2"/><span class="lab"><font color="Green" title="E67">A122</font></span>: 
<font color="Maroon" title="c10">k1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:33_3_1_1_2"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c9">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_3"><span class="lab"><font color="Green" title="E41">A91</font></span></a>, <a class="txt" href="#E17:33_3"><span class="lab"><font color="Green" title="E46">A96</font></span></a>, <a class="txt" href="#E29:33_3"><span class="lab"><font color="Green" title="E53">A103</font></span></a>, <a class="txt" href="#E32:33_3"><span class="lab"><font color="Green" title="E56">A106</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:33_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A123</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/><a NAME="E4:33_3_1_1_2"/>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_3_1_1_2"><span class="lab"><font color="Green" title="E67">A122</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E5:33_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A124</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:33_3"><span class="lab"><font color="Green" title="E59">A109</font></span></a>, <a class="txt" href="#E28:33_3"><span class="lab"><font color="Green" title="E52">A102</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E6:33_3_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">s1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c16">s2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">r1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_3"><span class="lab"><font color="Green" title="E39">A89</font></span></a>, <a class="txt" href="#E36:33_3"><span class="lab"><font color="Green" title="E58">A108</font></span></a>, <a class="txt" href="#E27:33_3"><span class="lab"><font color="Green" title="E51">A101</font></span></a>, <a class="ref" href="afinsq_1.html#T58" title="AFINSQ_1:th.58">AFINSQ_1:58</a></span>;<br/><a NAME="E7:33_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A125</font></span>: 
<font color="Maroon" title="c15">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">r1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T57" title="AFINSQ_1:th.57">AFINSQ_1:57</a></span>;<br/><a NAME="E8:33_3_1_1_2"/><span class="lab"><font color="Green" title="E71">A126</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">k</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"> ( <font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_3_1_1_2_1"/><span class="lab"><font color="Green" title="E72">A127</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 ; <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="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font></span><br/>

<a NAME="E2:33_3_1_1_2_1"/><span class="lab"><font color="Green" title="E73">A128</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:33_3"><span class="lab"><font color="Green" title="E53">A103</font></span></a>, <a class="txt" href="#E1:33_3_1_1_2_1"><span class="lab"><font color="Green" title="E72">A127</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E3:33_3_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A129</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_3"><span class="lab"><font color="Green" title="E41">A91</font></span></a>, <a class="txt" href="#E17:33_3"><span class="lab"><font color="Green" title="E46">A96</font></span></a>, <a class="txt" href="#E3:33_3_1_1_2"><span class="lab"><font color="Green" title="E68">A123</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:33_3_1_1_2_1"/>
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_3_1_1_2"><span class="lab"><font color="Green" title="E68">A123</font></span></a>, <a class="txt" href="#E2:33_3_1_1_2_1"><span class="lab"><font color="Green" title="E73">A128</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:33_3_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A130</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:33_3"><span class="lab"><font color="Green" title="E53">A103</font></span></a>, <a class="ref" href="funct_1.html#T48" title="FUNCT_1:th.48">FUNCT_1:48</a></span>;<br/>
<a NAME="E6:33_3_1_1_2_1"/><span class="lab"><font color="Green" title="E76">A131</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:33_3"><span class="lab"><font color="Green" title="E53">A103</font></span></a>, <a class="txt" href="#E5:33_3_1_1_2"><span class="lab"><font color="Green" title="E69">A124</font></span></a>, <a class="txt" href="#E1:33_3_1_1_2_1"><span class="lab"><font color="Green" title="E72">A127</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:33_3_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A132</font></span>: 
<font color="Maroon" title="c19">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:33_3"><span class="lab"><font color="Green" title="E59">A109</font></span></a>, <a class="txt" href="#E40:33_3"><span class="lab"><font color="Green" title="E62">A112</font></span></a></span>;<br/>
<a NAME="E8:33_3_1_1_2_1"/>
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_3_1_1_2_1"><span class="lab"><font color="Green" title="E76">A131</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E9:33_3_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">r1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_3_1_1_2_1"><span class="lab"><font color="Green" title="E73">A128</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E10:33_3_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A133</font></span>: 
<font color="Maroon" title="c19">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">r1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T48" title="FUNCT_1:th.48">FUNCT_1:48</a></span>;<br/>
<a NAME="E11:33_3_1_1_2_1"/>
<font color="Maroon" title="c15">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:33_3"><span class="lab"><font color="Green" title="E52">A102</font></span></a>, <a class="txt" href="#E29:33_3"><span class="lab"><font color="Green" title="E53">A103</font></span></a>, <a class="txt" href="#E31:33_3"><span class="lab"><font color="Green" title="E55">A105</font></span></a>, <a class="txt" href="#E1:33_3_1_1_2_1"><span class="lab"><font color="Green" title="E72">A127</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:33_3_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:33_3"><span class="lab"><font color="Green" title="E52">A102</font></span></a>, <a class="txt" href="#E7:33_3_1_1_2"><span class="lab"><font color="Green" title="E70">A125</font></span></a>, <a class="txt" href="#E7:33_3_1_1_2_1"><span class="lab"><font color="Green" title="E77">A132</font></span></a>, <a class="txt" href="#E10:33_3_1_1_2_1"><span class="lab"><font color="Green" title="E78">A133</font></span></a>, <a class="txt" href="#E3:33_3_1_1_2_1"><span class="lab"><font color="Green" title="E74">A129</font></span></a>, <a class="txt" href="#E5:33_3_1_1_2_1"><span class="lab"><font color="Green" title="E75">A130</font></span></a>, <a class="ref" href="funct_1.html#T48" title="FUNCT_1:th.48">FUNCT_1:48</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="E9:33_3_1_1_2"/>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:33_3"><span class="lab"><font color="Green" title="E49">A99</font></span></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></span>;<br/><a NAME="E10:33_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A134</font></span>: 
 ex <font color="Olive" title="b1">N</font> being    <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>  st <br/>( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">N</font> &amp; 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="c11">q</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">N</font> )
 ;<br/><a NAME="E11:33_3_1_1_2"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <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>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:33_3"><span class="lab"><font color="Green" title="E53">A103</font></span></a>, <a class="txt" href="#E3:33_3_1_1_2"><span class="lab"><font color="Green" title="E68">A123</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/><a NAME="E12:33_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <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="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_3_1_1_2"><span class="lab"><font color="Green" title="E71">A126</font></span></a>, <a class="ref" href="afinsq_1.html#T9" title="AFINSQ_1:th.9">AFINSQ_1:9</a></span>;<br/><a NAME="E13:33_3_1_1_2"/><span class="kw">then </span>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:33_3"><span class="lab"><font color="Green" title="E52">A102</font></span></a>, <a class="txt" href="#E29:33_3"><span class="lab"><font color="Green" title="E53">A103</font></span></a>, <a class="txt" href="#E10:33_3_1_1_2"><span class="lab"><font color="Green" title="E72">A134</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:33_3_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:33_3"><span class="lab"><font color="Green" title="E44">A94</font></span></a>, <a class="txt" href="#E37:33_3"><span class="lab"><font color="Green" title="E59">A109</font></span></a>, <a class="txt" href="#E29:33_3"><span class="lab"><font color="Green" title="E53">A103</font></span></a>, <a class="txt" href="#E5:33_3_1_1_2"><span class="lab"><font color="Green" title="E69">A124</font></span></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:33_3_1_1_3"/><span class="lab"><font color="Green" title="E67">A135</font></span>: 
<font color="Maroon" title="c10">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:33_3_1_1_3"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c11">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_3"><span class="lab"><font color="Green" title="E41">A91</font></span></a>, <a class="txt" href="#E38:33_3"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="txt" href="#E17:33_3"><span class="lab"><font color="Green" title="E46">A96</font></span></a>, <a class="txt" href="#E45:33_3"><span class="lab"><font color="Green" title="E65">A115</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:33_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A136</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>;<br/><a NAME="E4:33_3_1_1_3"/>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c12">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_3_1_1_3"><span class="lab"><font color="Green" title="E67">A135</font></span></a>, <a class="ref" href="xreal_1.html#T68" title="XREAL_1:th.68">XREAL_1:68</a></span>;<br/><a NAME="E5:33_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A137</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:33_3"><span class="lab"><font color="Green" title="E59">A109</font></span></a>, <a class="txt" href="#E28:33_3"><span class="lab"><font color="Green" title="E52">A102</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E6:33_3_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">r1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c20">r2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_3"><span class="lab"><font color="Green" title="E39">A89</font></span></a>, <a class="txt" href="#E36:33_3"><span class="lab"><font color="Green" title="E58">A108</font></span></a>, <a class="txt" href="#E27:33_3"><span class="lab"><font color="Green" title="E51">A101</font></span></a>, <a class="ref" href="afinsq_1.html#T58" title="AFINSQ_1:th.58">AFINSQ_1:58</a></span>;<br/><a NAME="E7:33_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A138</font></span>: 
<font color="Maroon" title="c19">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T57" title="AFINSQ_1:th.57">AFINSQ_1:57</a></span>;<br/><a NAME="E8:33_3_1_1_3"/><span class="lab"><font color="Green" title="E71">A139</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">k</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"> ( <font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_3_1_1_3_1"/><span class="lab"><font color="Green" title="E72">A140</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 ; <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="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font></span><br/>

<a NAME="E2:33_3_1_1_3_1"/><span class="lab"><font color="Green" title="E73">A141</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:33_3"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="txt" href="#E1:33_3_1_1_3_1"><span class="lab"><font color="Green" title="E72">A140</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E3:33_3_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A142</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33_3"><span class="lab"><font color="Green" title="E41">A91</font></span></a>, <a class="txt" href="#E17:33_3"><span class="lab"><font color="Green" title="E46">A96</font></span></a>, <a class="txt" href="#E3:33_3_1_1_3"><span class="lab"><font color="Green" title="E68">A136</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:33_3_1_1_3_1"/>
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_3_1_1_3"><span class="lab"><font color="Green" title="E68">A136</font></span></a>, <a class="txt" href="#E2:33_3_1_1_3_1"><span class="lab"><font color="Green" title="E73">A141</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:33_3_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A143</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:33_3"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="ref" href="funct_1.html#T48" title="FUNCT_1:th.48">FUNCT_1:48</a></span>;<br/>
<a NAME="E6:33_3_1_1_3_1"/><span class="lab"><font color="Green" title="E76">A144</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c15">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:33_3"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="txt" href="#E5:33_3_1_1_3"><span class="lab"><font color="Green" title="E69">A137</font></span></a>, <a class="txt" href="#E1:33_3_1_1_3_1"><span class="lab"><font color="Green" title="E72">A140</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:33_3_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A145</font></span>: 
<font color="Maroon" title="c15">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:33_3"><span class="lab"><font color="Green" title="E52">A102</font></span></a>, <a class="txt" href="#E31:33_3"><span class="lab"><font color="Green" title="E55">A105</font></span></a></span>;<br/>
<a NAME="E8:33_3_1_1_3_1"/>
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_3_1_1_3_1"><span class="lab"><font color="Green" title="E76">A144</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E9:33_3_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">s1</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_3_1_1_3_1"><span class="lab"><font color="Green" title="E73">A141</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E10:33_3_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A146</font></span>: 
<font color="Maroon" title="c15">s1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">s1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c19">r1</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T48" title="FUNCT_1:th.48">FUNCT_1:48</a></span>;<br/>
<a NAME="E11:33_3_1_1_3_1"/>
<font color="Maroon" title="c19">r1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:33_3"><span class="lab"><font color="Green" title="E59">A109</font></span></a>, <a class="txt" href="#E38:33_3"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="txt" href="#E40:33_3"><span class="lab"><font color="Green" title="E62">A112</font></span></a>, <a class="txt" href="#E1:33_3_1_1_3_1"><span class="lab"><font color="Green" title="E72">A140</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:33_3_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c22">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:33_3"><span class="lab"><font color="Green" title="E59">A109</font></span></a>, <a class="txt" href="#E7:33_3_1_1_3"><span class="lab"><font color="Green" title="E70">A138</font></span></a>, <a class="txt" href="#E7:33_3_1_1_3_1"><span class="lab"><font color="Green" title="E77">A145</font></span></a>, <a class="txt" href="#E10:33_3_1_1_3_1"><span class="lab"><font color="Green" title="E78">A146</font></span></a>, <a class="txt" href="#E3:33_3_1_1_3_1"><span class="lab"><font color="Green" title="E74">A142</font></span></a>, <a class="txt" href="#E5:33_3_1_1_3_1"><span class="lab"><font color="Green" title="E75">A143</font></span></a>, <a class="ref" href="funct_1.html#T48" title="FUNCT_1:th.48">FUNCT_1:48</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="E9:33_3_1_1_3"/>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:33_3"><span class="lab"><font color="Green" title="E44">A94</font></span></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></span>;<br/><a NAME="E10:33_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A147</font></span>: 
 ex <font color="Olive" title="b1">N</font> being    <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>  st <br/>( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">N</font> &amp; 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="c9">p</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">N</font> )
 ;<br/><a NAME="E11:33_3_1_1_3"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <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>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:33_3"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="txt" href="#E3:33_3_1_1_3"><span class="lab"><font color="Green" title="E68">A136</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/><a NAME="E12:33_3_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <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="c11">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_3_1_1_3"><span class="lab"><font color="Green" title="E71">A139</font></span></a>, <a class="ref" href="afinsq_1.html#T9" title="AFINSQ_1:th.9">AFINSQ_1:9</a></span>;<br/><a NAME="E13:33_3_1_1_3"/><span class="kw">then </span>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c10">k1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:33_3"><span class="lab"><font color="Green" title="E59">A109</font></span></a>, <a class="txt" href="#E38:33_3"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="txt" href="#E10:33_3_1_1_3"><span class="lab"><font color="Green" title="E72">A147</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:33_3_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:33_3"><span class="lab"><font color="Green" title="E60">A110</font></span></a>, <a class="txt" href="#E23:33_3"><span class="lab"><font color="Green" title="E49">A99</font></span></a>, <a class="txt" href="#E28:33_3"><span class="lab"><font color="Green" title="E52">A102</font></span></a>, <a class="txt" href="#E5:33_3_1_1_3"><span class="lab"><font color="Green" title="E69">A137</font></span></a>, <a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E48:33_3"/>
contradiction
 ; <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="E16:33"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>)</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="E17:33"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>)</span>)</span>,<font color="Maroon" title="c3">W</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="#E15:33"><span class="lab"><font color="Green" title="E36">A86</font></span></a>, <a class="txt" href="#E14:33"><span class="lab"><font color="Green" title="E35">A85</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="E18:33"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c2">n</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">m</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>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> (<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">m</font>,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</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/>


</div>
