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

<b>let </b><font color="Maroon" title="c1">seq</font>, <font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">seq2</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:52"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c1">seq</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a>
 <b>and </b><br/><a NAME="E2:52"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c2">seq1</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be    <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> ;<br/><a NAME="E1:52_1"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c2">seq1</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 <b>by </b><i><a class="txt" href="comseq_1.html#E2:52"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="comseq_1.html#T4" target="_self" title="COMSEQ_1:th.4">Th4</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="comseq_1.html#K8" title="COMSEQ_1:func.8">/"</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a> </span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_1.html#K7" title="COMSEQ_1:func.7">"</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D5" target="_self" title="COMSEQ_1:def.5">Def5</a>, <a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span></span>)</span></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_1.html#K7" title="COMSEQ_1:func.7">"</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="txt" href="comseq_1.html#E1:52_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="complex1.html#D7" title="COMPLEX1:def.7">COMPLEX1:def 7</a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_1.html#K7" title="COMSEQ_1:func.7">"</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_1.html#K7" title="COMSEQ_1:func.7">"</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D5" target="_self" title="COMSEQ_1:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="comseq_1.html#K7" title="COMSEQ_1:func.7">"</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_1.html#K7" title="COMSEQ_1:func.7">"</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D11" target="_self" title="COMSEQ_1:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_1.html#K7" title="COMSEQ_1:func.7">"</a> </span>)</span> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="comseq_1.html#K7" title="COMSEQ_1:func.7">"</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D5" target="_self" title="COMSEQ_1:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="comseq_1.html#K7" title="COMSEQ_1:func.7">"</a> </span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="txt" href="comseq_1.html#E1:52"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="comseq_1.html#E2:52"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="comseq_1.html#T33" target="_self" title="COMSEQ_1:th.33">Th33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="comseq_1.html#K8" title="COMSEQ_1:func.8">/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span></span>)</span> <a href="comseq_1.html#K1" title="COMSEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font>
<b>by </b><i><a class="ref" href="comseq_1.html#D5" target="_self" title="COMSEQ_1:def.5">Def5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:52"/>
<font color="Maroon" title="c3">seq2</font> <a href="comseq_1.html#K8" title="COMSEQ_1:func.8">/"</a> <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">seq2</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="comseq_1.html#K8" title="COMSEQ_1:func.8">/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="comseq_1.html#K3" title="COMSEQ_1:func.3">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span>
 <b>by </b><i><a class="ref" href="comseq_1.html#T6" target="_self" title="COMSEQ_1:th.6">Th6</a></i>;<br/>


</div>
