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<b>let </b><font color="Maroon" title="c1">seq1</font>, <font color="Maroon" title="c2">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c1">seq1</font> is    <a href="seqm_3.html#M1" title="SEQM_3:mode.1">subsequence</a> of <font color="Maroon" title="c2">seq</font> &amp; <font color="Maroon" title="c2">seq</font> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">Ns</font> being  <a href="seqm_3.html#V1" title="SEQM_3:attr.1">increasing</a> <a href="seqm_3.html#NM1" title="SEQM_3:NM.1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E3:10"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c1">seq1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">seq</font> <a href="seqm_3.html#K3" title="SEQM_3:func.3">*</a> <font color="Maroon" title="c3">Ns</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D10" title="SEQM_3:def.10">SEQM_3:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c4">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E2:10_1"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:10_1"/>
<font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">Ns</font> <a href="seqm_3.html#K2" title="SEQM_3:func.2">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="rfunct_2.html#E3:10"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="rfunct_2.html#E2:10_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="seqm_3.html#T31" title="SEQM_3:th.31">SEQM_3:31</a></i>;<br/><b>hence </b><a NAME="E4:10_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rfunct_2.html#E1:10"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:10"/>
<font color="Maroon" title="c1">seq1</font> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></i>;<br/>


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