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<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">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">B</font> holds <br/> <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> (<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">B</font>,<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>,<font color="Maroon" title="c1">n</font>,1,<font color="Maroon" title="c2">k</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">B</font> implies  <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> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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>,<font color="Maroon" title="c1">n</font>,1,<font color="Maroon" title="c2">k</font>)</span>)</span> )</span><br/>







<span class="kw">assume </span><a NAME="E1:22"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">B</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">  <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> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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>,<font color="Maroon" title="c1">n</font>,1,<font color="Maroon" title="c2">k</font>)</span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">T</font> = (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">W</font> = (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>);<br/>
<a NAME="E2:22"/><span class="lab"><font color="Green" title="E20">A2</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="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T62" onmouseover="rs('funct_4/T62')" onmouseout="rh()">FUNCT_4:62</a></span>;<br/>
<a NAME="E3:22"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="funct_4.html#T64" onmouseover="rs('funct_4/T64')" onmouseout="rh()">FUNCT_4:64</a></span>;<br/>
<a NAME="E4:22"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T64" onmouseover="rs('funct_4/T64')" onmouseout="rh()">FUNCT_4:64</a></span>;<br/>
<a NAME="E5:22"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T62" onmouseover="rs('funct_4/T62')" onmouseout="rh()">FUNCT_4:62</a></span>;<br/>
<a NAME="E6:22"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon" title="c3">A</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<a NAME="E7:22"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<font color="Maroon" title="c4">B</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<a NAME="E8:22"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="funcop_1.html#T88" onmouseover="rs('funcop_1/T88')" onmouseout="rh()">FUNCOP_1:88</a></span>;<br/>
<a NAME="E9:22"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="funcop_1.html#T88" onmouseover="rs('funcop_1/T88')" onmouseout="rh()">FUNCOP_1:88</a></span>;<br/>
<a NAME="E10:22"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="funcop_1.html#T88" onmouseover="rs('funcop_1/T88')" onmouseout="rh()">FUNCOP_1:88</a></span>;<br/>
<a NAME="E11:22"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default">1 <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="funcop_1.html#T88" onmouseover="rs('funcop_1/T88')" onmouseout="rh()">FUNCOP_1:88</a></span>;<br/>
<a NAME="E12:22"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> 1</span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E9:22"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#T11" onmouseover="rs('zfmisc_1/T11')" onmouseout="rh()">ZFMISC_1:11</a>, <a class="ref" href="funct_4.html#T92" onmouseover="rs('funct_4/T92')" onmouseout="rh()">FUNCT_4:92</a></span>;<br/>
<a NAME="E13:22"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default">1 <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c4">B</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:22"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T11" onmouseover="rs('zfmisc_1/T11')" onmouseout="rh()">ZFMISC_1:11</a>, <a class="ref" href="funct_4.html#T92" onmouseover="rs('funct_4/T92')" onmouseout="rh()">FUNCT_4:92</a></span>;<br/>
<a NAME="E14:22"/><span class="lab"><font color="Green" title="E32">A14</font></span>: 
(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="ref" href="funct_4.html#D4" onmouseover="rs('funct_4/D4')" onmouseout="rh()">FUNCT_4:def 4</a></span>;<br/>
<a NAME="E15:22"/><span class="lab"><font color="Green" title="E33">A15</font></span>: 
(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="funct_4.html#D4" onmouseover="rs('funct_4/D4')" onmouseout="rh()">FUNCT_4:def 4</a></span>;<br/>
<a NAME="E16:22"/><span class="lab"><font color="Green" title="E34">A16</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">A</font> <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="txt" href="#E1:22"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="funct_4.html#T63" onmouseover="rs('funct_4/T63')" onmouseout="rh()">FUNCT_4:63</a></span>;<br/>
<a NAME="E17:22"/><span class="lab"><font color="Green" title="E35">A17</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T63" onmouseover="rs('funct_4/T63')" onmouseout="rh()">FUNCT_4:63</a></span>;<br/>
<a NAME="E18:22"/><span class="lab"><font color="Green" title="E36">A18</font></span>: 
<span class="p1">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T63" onmouseover="rs('funct_4/T63')" onmouseout="rh()">FUNCT_4:63</a></span>;<br/>
<a NAME="E19:22"/><span class="lab"><font color="Green" title="E37">A19</font></span>: 
<span class="p1">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T63" onmouseover="rs('funct_4/T63')" onmouseout="rh()">FUNCT_4:63</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">f</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><span class="p1">(<span class="default">(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2;<br/>
<a NAME="E20:22"/><span class="lab"><font color="Green" title="E38">A20</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="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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>,<font color="Maroon" title="c1">n</font>,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="22_1"><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">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="c7">x</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> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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>,<font color="Maroon" title="c1">n</font>,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><br/>

<span class="kw">assume </span><a NAME="E1:22_1"/><span class="lab"><font color="Green" title="E39">A21</font></span>: 
<font color="Maroon" title="c7">x</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> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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>,<font color="Maroon" title="c1">n</font>,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><br/>

<a NAME="E2:22_1"/>
<font color="Maroon" title="c7">x</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> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E39">A21</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <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> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ;<br/>
<a NAME="E4:22_1"/><span class="lab"><font color="Green" title="E40">A22</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E5:22_1"/><span class="lab"><font color="Green" title="E41">A23</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:22_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
;<br/>
<a NAME="E7:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A24</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_1"><span class="lab"><font color="Green" title="E41">A23</font></span></a>, <a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E8:22_1"/><span class="lab"><font color="Green" title="E43">A25</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c8">f</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#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">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="relat_1.html#T26" onmouseover="rs('relat_1/T26')" onmouseout="rh()">RELAT_1:26</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">Tf</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c8">f</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">Tf</font> = <font color="Maroon" title="c9">Tf</font> as    <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">by</span> <span class="lab"><a class="txt" href="#E8:22_1"><span class="lab"><font color="Green" title="E43">A25</font></span></a>, <a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E11:22_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">Tf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_1"><span class="lab"><font color="Green" title="E42">A24</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">Tf</font> = <font color="Maroon" title="c10">Tf</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <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> <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="ref" href="finseq_2.html#T92" onmouseover="rs('finseq_2/T92')" onmouseout="rh()">FINSEQ_2:92</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">Tf</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="c11">Tf</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>,<font color="Maroon" title="c1">n</font>,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="Maroon" title="c11">Tf</font>] )</span><br/>

<a NAME="E13:22_1"/>
<font color="Maroon" title="c11">Tf</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,<font color="Maroon" title="c1">n</font>,1,<font color="Maroon" title="c2">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:22_1_1"/><span class="lab"><font color="Green" title="E44">A26</font></span>: 
<font color="Maroon" title="c3">A</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="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E2:22_1_1"/><span class="lab"><font color="Green" title="E45">A27</font></span>: 
<font color="Maroon" title="c4">B</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="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E3:22_1_1"/>
 <a href="relat_1.html#K10" title="RELAT_1:func.10">Coim</a> (<font color="Maroon" title="c11">Tf</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K10" title="RELAT_1:func.10">Coim</a> (<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c3">A</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1_1"><span class="lab"><font color="Green" title="E44">A26</font></span></a>, <a class="ref" href="ballot_1.html#T7" target="_self" onmouseover="rs('ballot_1/T7')" onmouseout="rh()">Th7</a>, <a class="txt" href="#E14:22"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="#E16:22"><span class="lab"><font color="Green" title="E34">A16</font></span></a></span>;<br/>
<a NAME="E4:22_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K10" title="RELAT_1:func.10">Coim</a> (<font color="Maroon" title="c11">Tf</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="ref" href="ballot_1.html#D1" target="_self" onmouseover="rs('ballot_1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:22_1_1"/>
<font color="Maroon" title="c11">Tf</font> <a href="tarski.html#R2" title="TARSKI: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>,<font color="Maroon" title="c1">n</font>,1,<font color="Maroon" title="c2">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ballot_1.html#D1" target="_self" onmouseover="rs('ballot_1/D1')" onmouseout="rh()">Def1</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ballot_1.html#D2" target="_self" onmouseover="rs('ballot_1/D2')" onmouseout="rh()">BALLOT_1:def 2</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">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Tf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Tf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c12">i</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"> ( <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Tf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Tf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E6:22_1_1"/><span class="lab"><font color="Green" title="E46">A28</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Tf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Tf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/>

<a NAME="E7:22_1_1"/><span class="lab"><font color="Green" title="E47">A29</font></span>: 
<font color="Maroon" title="c11">Tf</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T83" onmouseover="rs('relat_1/T83')" onmouseout="rh()">RELAT_1:83</a></span>;<br/>
<a NAME="E8:22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A30</font></span>: 
 <a href="relat_1.html#K10" title="RELAT_1:func.10">Coim</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">Tf</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K10" title="RELAT_1:func.10">Coim</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span>,<font color="Maroon" title="c3">A</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1_1"><span class="lab"><font color="Green" title="E44">A26</font></span></a>, <a class="ref" href="ballot_1.html#T7" target="_self" onmouseover="rs('ballot_1/T7')" onmouseout="rh()">Th7</a>, <a class="txt" href="#E14:22"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="#E16:22"><span class="lab"><font color="Green" title="E34">A16</font></span></a></span>;<br/>
<a NAME="E9:22_1_1"/><span class="lab"><font color="Green" title="E49">A31</font></span>: 
 <a href="relat_1.html#K10" title="RELAT_1:func.10">Coim</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">Tf</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K10" title="RELAT_1:func.10">Coim</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c12">i</font></span>)</span>,<font color="Maroon" title="c4">B</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_1_1"><span class="lab"><font color="Green" title="E47">A29</font></span></a>, <a class="txt" href="#E2:22_1_1"><span class="lab"><font color="Green" title="E45">A27</font></span></a>, <a class="ref" href="ballot_1.html#T7" target="_self" onmouseover="rs('ballot_1/T7')" onmouseout="rh()">Th7</a>, <a class="txt" href="#E14:22"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="#E17:22"><span class="lab"><font color="Green" title="E35">A17</font></span></a></span>;<br/>
<a NAME="E10:22_1_1"/>
<font color="Maroon" title="c8">f</font> is <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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="#E1:22_1"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="ref" href="ballot_1.html#D3" target="_self" onmouseover="rs('ballot_1/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:22_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Tf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Tf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1_1"><span class="lab"><font color="Green" title="E46">A28</font></span></a>, <a class="txt" href="#E8:22_1_1"><span class="lab"><font color="Green" title="E48">A30</font></span></a>, <a class="txt" href="#E9:22_1_1"><span class="lab"><font color="Green" title="E49">A31</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="E14:22_1"/>
( <font color="Maroon" title="c11">Tf</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>,<font color="Maroon" title="c1">n</font>,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="Maroon" title="c11">Tf</font>] )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ballot_1.html#D3" target="_self" onmouseover="rs('ballot_1/D3')" onmouseout="rh()">Def3</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="c7">C</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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>,<font color="Maroon" title="c1">n</font>,1,<font color="Maroon" title="c2">k</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E22:22"/><span class="lab"><font color="Green" title="E39">A32</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="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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="c7">C</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="#E20:22"><span class="lab"><font color="Green" title="E38">A20</font></span></a>);<br/></span>
<a NAME="E23:22"/>
 <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c1">n</font>,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="c7">C</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</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="c8">x</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>,<font color="Maroon" title="c1">n</font>,1,<font color="Maroon" title="c2">k</font>) or <font color="Maroon" title="c8">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="c7">C</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_2"/><span class="lab"><font color="Green" title="E40">A33</font></span>: 
<font color="Maroon" title="c8">x</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>,<font color="Maroon" title="c1">n</font>,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="c8">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="c7">C</font></span><br/>

<a NAME="E2:22_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</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>,<font color="Maroon" title="c1">n</font>,1,<font color="Maroon" title="c2">k</font>)
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <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> <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="E4:22_2"/><span class="lab"><font color="Green" title="E41">A34</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E5:22_2"/><span class="lab"><font color="Green" title="E42">A35</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:22_2"/><span class="lab"><font color="Green" title="E43">A36</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">f</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#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="E7:22_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A37</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_2"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E8:22_2"/><span class="lab"><font color="Green" title="E45">A38</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="relat_1.html#T26" onmouseover="rs('relat_1/T26')" onmouseout="rh()">RELAT_1:26</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">Wf</font> = <span class="p1">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">f</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="ref" href="finseq_1.html#D2" onmouseover="rs('finseq_1/D2')" onmouseout="rh()">FINSEQ_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">Wf</font> = <font color="Maroon" title="c10">Wf</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_2"><span class="lab"><font color="Green" title="E45">A38</font></span></a>, <a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E11:22_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Wf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">Wf</font> = <font color="Maroon" title="c11">Wf</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <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> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_2"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="ref" href="finseq_2.html#T92" onmouseover="rs('finseq_2/T92')" onmouseout="rh()">FINSEQ_2:92</a></span>;<br/>
<a NAME="E13:22_2"/>
<font color="Maroon" title="c12">Wf</font> is <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:22_2_1"/><span class="lab"><font color="Green" title="E46">A39</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E2:22_2_1"/><span class="lab"><font color="Green" title="E47">A40</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E3:22_2_1"/>
 <a href="relat_1.html#K10" title="RELAT_1:func.10">Coim</a> (<font color="Maroon" title="c12">Wf</font>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K10" title="RELAT_1:func.10">Coim</a> (<font color="Maroon" title="c9">f</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="ref" href="ballot_1.html#T7" target="_self" onmouseover="rs('ballot_1/T7')" onmouseout="rh()">Th7</a>, <a class="txt" href="#E15:22"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E36">A18</font></span></a></span>;<br/>
<a NAME="E4:22_2_1"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K10" title="RELAT_1:func.10">Coim</a> (<font color="Maroon" title="c12">Wf</font>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="ref" href="ballot_1.html#D1" target="_self" onmouseover="rs('ballot_1/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:22_2_1"/>
<font color="Maroon" title="c12">Wf</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ballot_1.html#K1" title="BALLOT_1:func.1">Election</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ballot_1.html#D1" target="_self" onmouseover="rs('ballot_1/D1')" onmouseout="rh()">Def1</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ballot_1.html#D2" target="_self" onmouseover="rs('ballot_1/D2')" onmouseout="rh()">BALLOT_1:def 2</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">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">Wf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">Wf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c13">i</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"> ( <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">Wf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">Wf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E6:22_2_1"/><span class="lab"><font color="Green" title="E48">A41</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">Wf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">Wf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/>

<a NAME="E7:22_2_1"/><span class="lab"><font color="Green" title="E49">A42</font></span>: 
<font color="Maroon" title="c12">Wf</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T83" onmouseover="rs('relat_1/T83')" onmouseout="rh()">RELAT_1:83</a></span>;<br/>
<a NAME="E8:22_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A43</font></span>: 
 <a href="relat_1.html#K10" title="RELAT_1:func.10">Coim</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">Wf</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c3">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K10" title="RELAT_1:func.10">Coim</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">i</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_2_1"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="ref" href="ballot_1.html#T7" target="_self" onmouseover="rs('ballot_1/T7')" onmouseout="rh()">Th7</a>, <a class="txt" href="#E15:22"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E36">A18</font></span></a></span>;<br/>
<a NAME="E9:22_2_1"/><span class="lab"><font color="Green" title="E51">A44</font></span>: 
 <a href="relat_1.html#K10" title="RELAT_1:func.10">Coim</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">Wf</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">i</font></span>)</span>,<font color="Maroon" title="c4">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K10" title="RELAT_1:func.10">Coim</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c13">i</font></span>)</span>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_2_1"><span class="lab"><font color="Green" title="E49">A42</font></span></a>, <a class="txt" href="#E2:22_2_1"><span class="lab"><font color="Green" title="E47">A40</font></span></a>, <a class="ref" href="ballot_1.html#T7" target="_self" onmouseover="rs('ballot_1/T7')" onmouseout="rh()">Th7</a>, <a class="txt" href="#E15:22"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E19:22"><span class="lab"><font color="Green" title="E37">A19</font></span></a></span>;<br/>
<a NAME="E10:22_2_1"/>
<font color="Maroon" title="c9">f</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,<font color="Maroon" title="c1">n</font>,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="#E1:22_2"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="ref" href="ballot_1.html#D3" target="_self" onmouseover="rs('ballot_1/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:22_2_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">Wf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">Wf</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c13">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_2_1"><span class="lab"><font color="Green" title="E48">A41</font></span></a>, <a class="txt" href="#E8:22_2_1"><span class="lab"><font color="Green" title="E50">A43</font></span></a>, <a class="txt" href="#E9:22_2_1"><span class="lab"><font color="Green" title="E51">A44</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>
<a NAME="E14:22_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A45</font></span>: 
<font color="Maroon" title="c12">Wf</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ballot_1.html#D3" target="_self" onmouseover="rs('ballot_1/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E15:22_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A46</font></span>: 
<font color="Maroon" title="c12">Wf</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">C</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>, <a class="txt" href="#E1:22_2"><span class="lab"><font color="Green" title="E40">A33</font></span></a></span>;<br/>
<font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">Wf</font> = 
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c12">Wf</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_2"><span class="lab"><font color="Green" title="E46">A45</font></span></a>, <a class="txt" href="#E22:22"><span class="lab"><font color="Green" title="E39">A32</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="necklace.html#T10" onmouseover="rs('necklace/T10')" onmouseout="rh()">NECKLACE:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>)</span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" onmouseover="rs('relat_1/T36')" onmouseout="rh()">RELAT_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K4" title="RELAT_1:func.4">id</a> <span class="p2"><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>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E5:22"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="funct_1.html#T39" onmouseover="rs('funct_1/T39')" onmouseout="rh()">FUNCT_1:39</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T53" onmouseover="rs('relat_1/T53')" onmouseout="rh()">RELAT_1:53</a>, <a class="txt" href="#E6:22_2"><span class="lab"><font color="Green" title="E43">A36</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E17:22_2"/>
<font color="Maroon" title="c8">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="c7">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:22_2"><span class="lab"><font color="Green" title="E47">A46</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="E24:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A47</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>,<font color="Maroon" title="c1">n</font>,1,<font color="Maroon" title="c2">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">C</font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3"><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="E1:22_3"/>
(  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c1">n</font>,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>  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>,<font color="Maroon" title="c1">n</font>,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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3_1"><span class="kw">suppose </span></a><a NAME="E1:22_3_1"/><span class="lab"><font color="Green" title="E41">A48</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>,<font color="Maroon" title="c1">n</font>,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#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> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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>,<font color="Maroon" title="c1">n</font>,1,<font color="Maroon" title="c2">k</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:22_3_1"/>
 <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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="22_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:22_3_1_1"/>
 <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:22_3_1_1"/><span class="lab"><font color="Green" title="E42">A49</font></span>: 
<font color="Maroon" title="c8">x</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> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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="E4:22_3_1_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>,<font color="Maroon" title="c1">n</font>,1,<font color="Maroon" title="c2">k</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:22"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="txt" href="#E3:22_3_1_1"><span class="lab"><font color="Green" title="E42">A49</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:22_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_3_1"><span class="lab"><font color="Green" title="E41">A48</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="E3:22_3_1"/>
 <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> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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>,<font color="Maroon" title="c1">n</font>,1,<font color="Maroon" title="c2">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_3_1"><span class="lab"><font color="Green" title="E41">A48</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3_2"><span class="kw">suppose </span></a><a NAME="E1:22_3_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>,<font color="Maroon" title="c1">n</font>,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#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> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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>,<font color="Maroon" title="c1">n</font>,1,<font color="Maroon" title="c2">k</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:22_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A50</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ballot_1.html#K2" title="BALLOT_1:func.2">DominatedElection</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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="E3:22_3_2"/>
<font color="Maroon" title="c7">C</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="22_3_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">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="c8">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="c7">C</font> or  not <font color="Maroon" title="c9">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="c7">C</font> or  not <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x2</font> or <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:22_3_2_1"/><span class="lab"><font color="Green" title="E42">A51</font></span>: 
<font color="Maroon" title="c8">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="c7">C</font>
 <span class="kw">and </span><br/><a NAME="E2:22_3_2_1"/><span class="lab"><font color="Green" title="E43">A52</font></span>: 
<font color="Maroon" title="c9">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="c7">C</font>
 <span class="kw">and </span><br/><a NAME="E3:22_3_2_1"/><span class="lab"><font color="Green" title="E44">A53</font></span>: 
<font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font></span><br/>

<a NAME="E4:22_3_2_1"/><span class="lab"><font color="Green" title="E45">A54</font></span>: 
<font color="Maroon" title="c8">x1</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> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_3_2"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="txt" href="#E1:22_3_2_1"><span class="lab"><font color="Green" title="E42">A51</font></span></a></span>;<br/>
<a NAME="E5:22_3_2_1"/>
<font color="Maroon" title="c9">x2</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> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c2">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_3_2"><span class="lab"><font color="Green" title="E41">A50</font></span></a>, <a class="txt" href="#E2:22_3_2_1"><span class="lab"><font color="Green" title="E43">A52</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">x1</font> = <font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c11">x2</font> = <font color="Maroon" title="c9">x2</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <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> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_3_2_1"><span class="lab"><font color="Green" title="E45">A54</font></span></a></span>;<br/>
<a NAME="E7:22_3_2_1"/><span class="lab"><font color="Green" title="E46">A55</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">x1</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E8:22_3_2_1"/><span class="lab"><font color="Green" title="E47">A56</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">x2</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E9:22_3_2_1"/><span class="lab"><font color="Green" title="E48">A57</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22_3_2_1"><span class="lab"><font color="Green" title="E46">A55</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E10:22_3_2_1"/><span class="lab"><font color="Green" title="E49">A58</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22_3_2_1"><span class="lab"><font color="Green" title="E47">A56</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:22_3_2_1"/><span class="lab"><font color="Green" title="E50">A59</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E20">A2</font></span></a></span>;<br/>
<a NAME="E12:22_3_2_1"/><span class="lab"><font color="Green" title="E51">A60</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22"><span class="lab"><font color="Green" title="E20">A2</font></span></a></span>;<br/>
<a NAME="E13:22_3_2_1"/><span class="lab"><font color="Green" title="E52">A61</font></span>: 
(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="ref" href="funct_4.html#D4" onmouseover="rs('funct_4/D4')" onmouseout="rh()">FUNCT_4:def 4</a></span>;<br/>
<a NAME="E14:22_3_2_1"/><span class="lab"><font color="Green" title="E53">A62</font></span>: 
<font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c10">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_3_2_1"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E22:22"><span class="lab"><font color="Green" title="E39">A32</font></span></a></span>;<br/>
<a NAME="E15:22_3_2_1"/>
<font color="Maroon" title="c7">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>) <a href="funct_4.html#K7" title="FUNCT_4:func.7">--&gt;</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">*</a> <font color="Maroon" title="c11">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:22_3_2_1"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E22:22"><span class="lab"><font color="Green" title="E39">A32</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:22_3_2_1"/>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22_3_2_1"><span class="lab"><font color="Green" title="E53">A62</font></span></a>, <a class="txt" href="#E9:22_3_2_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="txt" href="#E10:22_3_2_1"><span class="lab"><font color="Green" title="E49">A58</font></span></a>, <a class="txt" href="#E11:22_3_2_1"><span class="lab"><font color="Green" title="E50">A59</font></span></a>, <a class="txt" href="#E12:22_3_2_1"><span class="lab"><font color="Green" title="E51">A60</font></span></a>, <a class="txt" href="#E13:22_3_2_1"><span class="lab"><font color="Green" title="E52">A61</font></span></a>, <a class="txt" href="#E3:22_3_2_1"><span class="lab"><font color="Green" title="E44">A53</font></span></a>, <a class="ref" href="funct_1.html#T27" onmouseover="rs('funct_1/T27')" onmouseout="rh()">FUNCT_1:27</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="E4:22_3_2"/>
 <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> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c4">B</font>,<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>,<font color="Maroon" title="c1">n</font>,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="#E24:22"><span class="lab"><font color="Green" title="E40">A47</font></span></a>, <a class="txt" href="#E2:22_3_2"><span class="lab"><font color="Green" title="E41">A50</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><span class="kw">end;</span></div>

</div>
