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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>

<a NAME="E1:1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="comseq_2.html#D2">COMSEQ_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="comseq_2.html#D2">COMSEQ_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D15">COMPLEX1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T112">COMPLEX1:112</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T112">COMPLEX1:112</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>

;<br/>
<b>hence </b><a NAME="E4:1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_2.html#K1">*'</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


</div>
