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<b>let </b><font color="Maroon">seq1</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:8"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">seq1</font> is    <a href="comseq_3.html#M1">subsequence</a> of <font color="Maroon">seq</font> &amp; <font color="Maroon">seq</font> is <a href="comseq_1.html#V1">non-zero</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">Ns</font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E3:8"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">seq1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">seq</font> <a href="comseq_3.html#K5">*</a> <font color="Maroon">Ns</font>
 <b>by </b><i><a class="ref" href="comseq_3.html#D9">COMSEQ_3:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:8_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">seq1</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 ;<br/><a NAME="E2:8_1"/>
<font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ns</font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><b>hence </b><a NAME="E3:8_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="comseq_1.html#T4">COMSEQ_1:4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:8"/>
<font color="Maroon">seq1</font> is <a href="comseq_1.html#V1">non-zero</a>
 <b>by </b><i><a class="ref" href="comseq_1.html#T4">COMSEQ_1:4</a></i>;<br/>


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