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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">R</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:3"/><i><font color="Green" title="E4">A1</font></i>: 
(  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">R</font> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">R</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) )
 ;<br/>

<b>given </b><font color="Maroon" title="c3">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E3:3"/><i><font color="Green" title="E5">A2</font></i>: 
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c2">R</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E4:3"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">R</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">k</font>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="rfunct_4.html#E3:3"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:3"/>
contradiction
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="rfunct_4.html#E3:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="rvsum_1.html#T115" title="RVSUM_1:th.115">RVSUM_1:115</a></i>;<br/>


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