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<div class="add">

<span class="kw">set </span><font color="Maroon" title="c5">BALL</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">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="c1">A</font>,<font color="Maroon" title="c3">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : <font color="Olive" title="b1">v</font> is <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</a>  </span> } </span> ;<br/>
<a NAME="E1:19_1_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">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="c1">A</font>,<font color="Maroon" title="c3">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : <font color="Olive" title="b1">v</font> is <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</a>  </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ballot_1.html#K1" title="BALLOT_1:func.1">Election</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">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="c1">A</font>,<font color="Maroon" title="c3">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : <font color="Olive" title="b1">v</font> is <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</a>  </span> } </span>  or <font color="Maroon" title="c6">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="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:19_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">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="c1">A</font>,<font color="Maroon" title="c3">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : <font color="Olive" title="b1">v</font> is <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</a>  </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">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="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font>)</span><br/>

<a NAME="E2:19_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">v</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">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="c1">A</font>,<font color="Maroon" title="c3">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> &amp; <font color="Olive" title="b1">v</font> is <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:19_1_1_1"/>
<font color="Maroon" title="c6">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="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">BALL</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">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="c1">A</font>,<font color="Maroon" title="c3">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> : <font color="Olive" title="b1">v</font> is <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</a>  </span> } </span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="ballot_1.html#K1" title="BALLOT_1:func.1">Election</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font>)</span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">BALL</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">  for <font color="Olive" title="b1">f</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">BALL</font> iff <font color="Olive" title="b1">f</font> is <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</a>  )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">f</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">BALL</font> iff <font color="Maroon" title="c7">f</font> is <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</a>  )</span><br/>

<a NAME="E3:19_1_1"/>
( <font color="Maroon" title="c7">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">BALL</font> implies <font color="Maroon" title="c7">f</font> is <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">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="19_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:19_1_1_2"/>
<font color="Maroon" title="c7">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">BALL</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f</font> is <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</a> </span><br/>

<a NAME="E2:19_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">v</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">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="c1">A</font>,<font color="Maroon" title="c3">B</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> st <br/>( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> &amp; <font color="Olive" title="b1">v</font> is <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:19_1_1_2"/>
<font color="Maroon" title="c7">f</font> is <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:19_1_1"/>
( <font color="Maroon" title="c7">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">BALL</font> iff <font color="Maroon" title="c7">f</font> is <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">k</font> <a href="ballot_1.html#V1" title="BALLOT_1:attr.1">-dominated-election</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"> verum</span><br/>


</div>
