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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> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon" title="c2">k</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="ballot_1.html#K1" title="BALLOT_1:func.1">Election</a> (<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">A</font>,<font color="Maroon" title="c2">k</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</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="c2">k</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="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="c3">A</font>,<font color="Maroon" title="c2">k</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:8"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c2">k</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="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="c3">A</font>,<font color="Maroon" title="c2">k</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/>

<span class="kw">assume </span><a NAME="E2:8"/>
 not  <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="c3">A</font>,<font color="Maroon" title="c2">k</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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="c4">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:8"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c4">v</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="c3">A</font>,<font color="Maroon" title="c2">k</font>)
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">v</font> = <font color="Maroon" title="c4">v</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#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="#E4:8"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="enumset1.html#T29" onmouseover="rs('enumset1/T29')" onmouseout="rh()">ENUMSET1:29</a></span>;<br/>
<a NAME="E6:8"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">v</font></span>)</span> <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="E7:8"/>
<font color="Maroon" title="c5">v</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">v</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="ref" href="relat_1.html#T143" onmouseover="rs('relat_1/T143')" onmouseout="rh()">RELAT_1:143</a></span>;<br/>
<a NAME="E8:8"/><span class="kw">then </span>
<font color="Maroon" title="c5">v</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T134" onmouseover="rs('relat_1/T134')" onmouseout="rh()">RELAT_1:134</a>, <a class="ref" href="relat_1.html#T132" onmouseover="rs('relat_1/T132')" onmouseout="rh()">RELAT_1:132</a></span>;<br/>
<a NAME="E9:8"/><span class="kw">then </span>
<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> <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="#E6:8"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="ballot_1.html#D1" target="_self" onmouseover="rs('ballot_1/D1')" onmouseout="rh()">Def1</a>, <a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E10">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E9">A1</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/>


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