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<b>let </b><font color="Maroon">p</font> be  <a href="rvsum_1.html#V1" target="_self">complex-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:112"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:112"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:112"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T63">FINSEQ_1:63</a></i>;<br/>
<a NAME="E5:112"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K2">COMPLEX</a> 
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E6:112"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">p</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E7:112"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">q</font> holds <br/><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K2">COMPLEX</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:112_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:112_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T18">FINSEQ_2:18</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:112_1"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K2">COMPLEX</a> 
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:112"/><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T33" target="_self">Th33</a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E9:112"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E10:112"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:112"/><b>then </b>
( <font color="Maroon">d</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  &amp; <font color="Maroon">q</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a>  )
 
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a>, <a class="ref" href="finseq_2.html#T14">FINSEQ_2:14</a></i>;<br/>
<b>hence </b><a NAME="E12:112"/>
 ex <font color="Olive">q</font> being <a href="rvsum_1.html#V1" target="_self">complex-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> ex <font color="Olive">d</font> being <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Olive">d</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i/>;<br/>


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