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

<b>consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E2:96_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp;  <a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="finsop_1.html#K2">$$</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="txt" href="rvsum_1.html#E5:96_1_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E3:96_1_1"/><i><font color="Green">E51</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E7:96_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="seq_1.html#D1">SEQ_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">mc</font> =  <a href="binop_2.html#K29">multcomplex</a> ;<br/>
<b>set </b><font color="Maroon">mr</font> =  <a href="binop_2.html#K35">multreal</a> ;<br/>
<a NAME="E5:96_1_1"/><i><font color="Green">E60</font></i>: 
<a href="binop_2.html#K29">multcomplex</a>  <a href="finsop_1.html#K2">$$</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="setwop_2.html#D2">SETWOP_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:96_1_1"/><i><font color="Green">E61</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><a href="binop_2.html#K29">multcomplex</a>  <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span></span>)</span> is <a href="xreal_0.html#V1">real</a>;<br/>
<a NAME="E8:96_1_1"/>
 <a href="setwop_2.html#K2">Seg</a> 0 <a href="hidden.html#R1">=</a>  <a href="setwiseo.html#K1">{}.</a> <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E9:96_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="setwiseo.html#T40">SETWISEO:40</a>, <a class="ref" href="binop_2.html#T6">BINOP_2:6</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> =  <a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span>;<br/>
<a NAME="E10:96_1_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:96_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:96_1_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<a href="binop_2.html#K29">multcomplex</a>  <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span></span>)</span> is <a href="xreal_0.html#V1">real</a>
 
;<br/>
<a NAME="E3:96_1_1_1"/><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> is <a href="xreal_0.html#V1">real</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_1_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font> or  not <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:96_1_1_1_1_1_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:96_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="setwop_2.html#T22">SETWOP_2:22</a></i>;<br/><a NAME="E3:96_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T33" target="_self">Th33</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E4:96_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> is <a href="xreal_0.html#V1">real</a>
 <b>by </b><i>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:96_1_1_1_1_1_2"/>
not <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:96_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="binop_2.html#T6">BINOP_2:6</a>, <a class="ref" href="setwop_2.html#T22">SETWOP_2:22</a></i>;<br/><b>hence </b><a NAME="E3:96_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> is <a href="xreal_0.html#V1">real</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>, <font color="Maroon">b</font> = <a href="binop_2.html#K29">multcomplex</a>  <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span></span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="rvsum_1.html#T27" target="_self">Th27</a>, <a class="ref" href="rvsum_1.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E5:96_1_1_1"/>
not <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T9">FINSEQ_3:9</a></i>;<br/>
<b>then </b><a href="binop_2.html#K29">multcomplex</a>  <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span></span>)</span> = 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="setwiseo.html#K7">$$</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setwop_2.html#K1">[#]</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="setwop_2.html#T4">SETWOP_2:4</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E7:96_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1]
 <b>by </b><i><a class="ref" href="finseq_1.html#T11">FINSEQ_1:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:96_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="rvsum_1.html#T26" target="_self">Th26</a>);<br/></i>
<a NAME="E12:96_1_1"/><b>then </b>
<a href="binop_2.html#K29">multcomplex</a>  <a href="finsop_1.html#K2">$$</a> <font color="Maroon">f</font> is <a href="xreal_0.html#V1">real</a>
 
<b>by </b><i>, <a class="txt" href="rvsum_1.html#E9:96_1_1"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:96_1_1"/>
 <a href="rvsum_1.html#K16" target="_self">Product</a> <font color="Maroon">F</font> is <a href="xreal_0.html#V1">real</a>
 <b>by </b><i><a class="txt" href="rvsum_1.html#E7:96_1_1"><i><font color="Green">E61</font></i></a></i>;<br/>


</div>
