<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:53"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="complex1.html#K5">0c</a> 
 <b>and </b><br/><a NAME="E2:53"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="comseq_1.html#V1" target="_self">non-zero</a>
 <b>and </b><br/><a NAME="E3:53"/><i><font color="Green">E42</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K4" target="_self">(#)</a> <font color="Maroon">c<sub>2</sub></font> is <a href="comseq_1.html#V1" target="_self">non-zero</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:53"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K4" target="_self">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 <b>by </b><i><a class="txt" href="comseq_1.html#E3:53"><i><font color="Green">E42</font></i></a>, <a class="ref" href="comseq_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E6:53"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="complex1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i><a class="txt" href="comseq_1.html#E5:53"><i><font color="Green">E43</font></i></a>, <a class="ref" href="comseq_1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E7:53"/>
<font color="Maroon">c<sub>2</sub></font> <a href="comseq_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></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#E2:53"><i><font color="Green">E41</font></i></a>, <a class="ref" href="comseq_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E8:53"/>
contradiction
 <b>by </b><i><a class="txt" href="comseq_1.html#E1:53"><i><font color="Green">E40</font></i></a>, <a class="txt" href="comseq_1.html#E6:53"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>


</div>
