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



<b>thus </b><a NAME="E1:45"/>
( <font color="Maroon">seq</font> is <a href="comseq_1.html#V1" target="_self">non-zero</a> &amp; <font color="Maroon">seq1</font> is <a href="comseq_1.html#V1" target="_self">non-zero</a> implies <font color="Maroon">seq</font> <a href="comseq_1.html#K3" target="_self">(#)</a> <font color="Maroon">seq1</font> is <a href="comseq_1.html#V1" target="_self">non-zero</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:45_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">seq</font> is <a href="comseq_1.html#V1" target="_self">non-zero</a>
 <b>and </b><br/><a NAME="E2:45_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">seq1</font> is <a href="comseq_1.html#V1" target="_self">non-zero</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:45_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">seq</font> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 <b>by </b><i><a class="txt" href="comseq_1.html#E2:45_1_1"><i><font color="Green">E35</font></i></a>, </i>;<br/><a NAME="E2:45_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">seq1</font> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 <b>by </b><i><a class="txt" href="comseq_1.html#E2:45"><i><font color="Green">E36</font></i></a>, </i>;<br/><a NAME="E3:45_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K3" target="_self">(#)</a> <font color="Maroon">seq1</font></span>)</span> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:45_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K3" target="_self">(#)</a> <font color="Maroon">seq1</font></span>)</span> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 <b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:45_1"/>
<font color="Maroon">seq</font> <a href="comseq_1.html#K3" target="_self">(#)</a> <font color="Maroon">seq1</font> is <a href="comseq_1.html#V1" target="_self">non-zero</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:45"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">seq</font> <a href="comseq_1.html#K3" target="_self">(#)</a> <font color="Maroon">seq1</font> is <a href="comseq_1.html#V1" target="_self">non-zero</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:45_2"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K3" target="_self">(#)</a> <font color="Maroon">seq1</font></span>)</span> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E2:45_2"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K3" target="_self">(#)</a> <font color="Maroon">seq1</font></span>)</span> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:45_2"/>
<font color="Maroon">seq</font> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 <b>by </b><i><a class="txt" href="comseq_1.html#E2:45"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:45"/>
<font color="Maroon">seq</font> is <a href="comseq_1.html#V1" target="_self">non-zero</a>
 <b>by </b><i/>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:45_3"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K3" target="_self">(#)</a> <font color="Maroon">seq1</font></span>)</span> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E2:45_3"/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K3" target="_self">(#)</a> <font color="Maroon">seq1</font></span>)</span> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:45_3"/>
<font color="Maroon">seq1</font> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 <b>by </b><i><a class="txt" href="comseq_1.html#E2:45"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:45"/>
<font color="Maroon">seq1</font> is <a href="comseq_1.html#V1" target="_self">non-zero</a>
 <b>by </b><i/>;<br/>


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