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

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

<b>reconsider </b><font color="Maroon">rseq1</font> = <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span> as   <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<b>reconsider </b><font color="Maroon">rseq2</font> = <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> as   <a href="seq_1.html#NM1">Real_Sequence</a> ;<br/>
<a NAME="E3:19"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </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/>

<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="comseq_2.html#D2">COMSEQ_2:def 2</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T139">COMPLEX1:139</a></i>
;<br/>
<b>hence </b><a NAME="E2:19_1"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="comseq_1.html#D14">COMSEQ_1:def 14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:19"/><b>then </b>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">rseq1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rseq2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font>
 
;<br/>
<b>hence </b><a NAME="E5:19"/>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">seq</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span><a href="comseq_1.html#K9">.|</a></span>
 <b>by </b><i><a class="ref" href="seq_1.html#T8">SEQ_1:8</a></i>;<br/>


</div>
