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

<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>)</span>)</span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">D</font> =  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">B</font> =  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">Z</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">d</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Olive" title="b1">d</font></span>)</span>;<br/>
<a NAME="E1:27"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>) holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>) implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</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="E1:27_1"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</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">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  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:27_1"/><span class="lab"><font color="Green" title="E25">A3</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="E4:27_1"/>
( <font color="Maroon" title="c7">p</font> is  <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font> &amp;  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="catalan2.html#D2" onmouseover="rs('catalan2/D2')" onmouseout="rh()">CATALAN2:def 2</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c8">y</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c7">p</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">y</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E5:27_1"/>
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">y</font>] )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ballot_1.html#T24" target="_self" onmouseover="rs('ballot_1/T24')" onmouseout="rh()">Th24</a>, <a class="txt" href="#E3:27_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:27"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>) 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" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E23">A1</font></span></a>);<br/></span>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:27_2"/>
(  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1"><span class="kw">suppose </span></a><a NAME="E1:27_2_1"/>
 <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:27_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A5</font></span>: 
 <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>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E3:27_2_1"/>
 <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>) or <font color="Maroon" title="c7">y</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="E1:27_2_1_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">y</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/>

<a NAME="E2:27_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ballot_1.html#K1" title="BALLOT_1:func.1">Election</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>)
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">y</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ;<br/>
<a NAME="E4:27_2_1_1"/>
<font color="Maroon" title="c8">g</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1,1,<font color="Maroon" title="c2">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_2_1_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="ballot_1.html#D3" target="_self" onmouseover="rs('ballot_1/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E5:27_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ballot_1.html#T23" target="_self" onmouseover="rs('ballot_1/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">d</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <font color="Maroon" title="c10">dg</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span><span class="kw"> such that </span><br/><a NAME="E7:27_2_1_1"/><span class="lab"><font color="Green" title="E28">A8</font></span>: 
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">and </span><br/><a NAME="E8:27_2_1_1"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">d</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">dg</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T102" onmouseover="rs('finseq_3/T102')" onmouseout="rh()">FINSEQ_3:102</a></span>;<br/>
<a NAME="E9:27_2_1_1"/>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
;<br/>
<a NAME="E10:27_2_1_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"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">FS2XFS</a> <font color="Maroon" title="c10">dg</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">G</font> =  <a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">FS2XFS</a> <font color="Maroon" title="c10">dg</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<a NAME="E12:27_2_1_1"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
 <a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c11">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">dg</font>
 
;<br/>
<a NAME="E13:27_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A11</font></span>: 
<font color="Maroon" title="c11">G</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_2_1_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E7:27_2_1_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="#E8:27_2_1_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E5:27_2_1_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="ballot_1.html#T24" target="_self" onmouseover="rs('ballot_1/T24')" onmouseout="rh()">Th24</a></span>;<br/>
<a NAME="E14:27_2_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:27_2_1_1"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="txt" href="#E1:27_2_1_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E7:27_2_1_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="#E8:27_2_1_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E5:27_2_1_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="ballot_1.html#T24" target="_self" onmouseover="rs('ballot_1/T24')" onmouseout="rh()">Th24</a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E24">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:27_2_1_1"/>
<font color="Maroon" title="c7">y</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="#E13:27_2_1_1"><span class="lab"><font color="Green" title="E31">A11</font></span></a>, <a class="txt" href="#E2:27_2_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E4:27_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A12</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>  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>)
 ;<br/><a NAME="E5:27_2_1"/>
<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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">f</font> or  not <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</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">x1</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">x2</font> or <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27_2_1_2"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<font color="Maroon" title="c7">x1</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="E2:27_2_1_2"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<font color="Maroon" title="c8">x2</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="E3:27_2_1_2"/><span class="lab"><font color="Green" title="E29">A15</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">x1</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">x2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">p1</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:27_2_1_2"/><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<font color="Maroon" title="c9">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font>
 <span class="kw">and </span><br/><a NAME="E6:27_2_1_2"/>
( <font color="Maroon" title="c9">p1</font> is  <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font> &amp;  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c9">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_2_1_2"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="ref" href="catalan2.html#D2" onmouseover="rs('catalan2/D2')" onmouseout="rh()">CATALAN2:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">p2</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E8:27_2_1_2"/><span class="lab"><font color="Green" title="E31">A17</font></span>: 
<font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font>
 <span class="kw">and </span><br/><a NAME="E9:27_2_1_2"/>
( <font color="Maroon" title="c10">p2</font> is  <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font> &amp;  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_2_1_2"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="ref" href="catalan2.html#D2" onmouseover="rs('catalan2/D2')" onmouseout="rh()">CATALAN2:def 2</a></span>;<br/>
<a NAME="E10:27_2_1_2"/><span class="lab"><font color="Green" title="E32">A18</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="c9">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c9">p1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_2_1_2"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E1:27_2_1_2"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E24">A4</font></span></a></span>;<br/>
<a NAME="E11:27_2_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c10">p2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_2_1_2"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E8:27_2_1_2"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E24">A4</font></span></a></span>;<br/>
<a NAME="E12:27_2_1_2"/><span class="kw">then </span>
 <a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c9">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">XFS2FS</a> <font color="Maroon" title="c10">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:27_2_1_2"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E3:27_2_1_2"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E5:27_2_1_2"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E8:27_2_1_2"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="finseq_1.html#T33" onmouseover="rs('finseq_1/T33')" onmouseout="rh()">FINSEQ_1:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:27_2_1_2"/>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ballot_1.html#T4" target="_self" onmouseover="rs('ballot_1/T4')" onmouseout="rh()">Th4</a>, <a class="txt" href="#E5:27_2_1_2"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E8:27_2_1_2"><span class="lab"><font color="Green" title="E31">A17</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">hence </span><a NAME="E6:27_2_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#D4" onmouseover="rs('wellord2/D4')" onmouseout="rh()">WELLORD2:def 4</a>, <a class="txt" href="#E2:27_2_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E4:27_2_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="card_1.html#T5" onmouseover="rs('card_1/T5')" onmouseout="rh()">CARD_1:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="27_2_2"><span class="kw">suppose </span></a><a NAME="E1:27_2_2"/><span class="lab"><font color="Green" title="E25">A19</font></span>: 
 <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:27_2_2"/>
 <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:27_2_2_1"/>
 <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:27_2_2_1"/><span class="lab"><font color="Green" title="E26">A20</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E4:27_2_2_1"/>
 ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_2_2_1"><span class="lab"><font color="Green" title="E26">A20</font></span></a>, <a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E23">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:27_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_2_2"><span class="lab"><font color="Green" title="E25">A19</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">hence </span><a NAME="E3:27_2_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K2" title="CATALAN2:func.2">Domin_0</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">k</font></span>)</span>,<font color="Maroon" title="c2">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1,<font color="Maroon" title="c2">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_2_2"><span class="lab"><font color="Green" title="E25">A19</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
