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

<b>set </b><font color="Maroon">mru</font> = the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E1:45_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="binop_2.html#K29">multcomplex</a>  <a href="funct_2.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<a NAME="E2:45_1_1"/>
<font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> 
 
;<br/>
<a NAME="E3:45_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K2">COMPLEX</a> 
 
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<a NAME="E4:45_1_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K2">COMPLEX</a> ,<a href="numbers.html#K2">COMPLEX</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E5:45_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <a href="binop_2.html#K29">multcomplex</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:45_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="binop_2.html#K29">multcomplex</a> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E7:45_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E8:45_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:45_1_1_1"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:45_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E4:45_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  )
 
<b>by </b><i><a class="txt" href="uniroots.html#E4:45_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="uniroots.html#T2" target="_self">Th2</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i>, <a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<a NAME="E7:45_1_1_1"/><i><font color="Green">E53</font></i>: 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="binop_2.html#K5">*</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>;<br/>
<a NAME="E8:45_1_1_1"/>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="uniroots.html#E3:45_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="uniroots.html#E4:45_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="uniroots.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E9:45_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="binop_2.html#K5">*</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:45_1_1_1"/>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 <b>by </b><i><a class="ref" href="uniroots.html#T2" target="_self">Th2</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">uM</font> = the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<a NAME="E10:45_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">uM</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>set </b><font color="Maroon">H</font> =  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">uM</font> #);<br/>
<a NAME="E11:45_1_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">r</font>, <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">uM</font> #) holds  <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="group_1.html#K1">*</a> <font color="Olive">s</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="group_1.html#K1">*</a> <font color="Olive">t</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">uM</font> #), <font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">uM</font> #), <font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">uM</font> #);<br/>

<a NAME="E1:45_1_1_2"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a>  &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a>  )
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E2:45_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  &amp; <font color="Maroon">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  &amp; <font color="Maroon">t</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  )
 
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">mc</font> =  <a href="binop_2.html#K29">multcomplex</a> ;<br/>
<a NAME="E3:45_1_1_2"/><i><font color="Green">E58</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:45_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
( <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E5:45_1_1_2"/>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E6:45_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:45_1_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">r</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,<font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="uniroots.html#E4"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E8:45_1_1_2"/>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E9:45_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:45_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">t</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="uniroots.html#E4"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E11:45_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="group_1.html#K1">*</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">1F</font> =  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">uM</font> #) <b>by </b><i/>;<br/>
<a NAME="E13:45_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> 0</span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> 0</span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
 
<b>by </b><i><a class="ref" href="complfld.html#T10">COMPLFLD:10</a>, <a class="ref" href="sin_cos.html#T34">SIN_COS:34</a></i>;<br/>
<a NAME="E14:45_1_1"/>
for <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">uM</font> #) holds <br/> ( <font color="Olive">s1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">1F</font> <a href="hidden.html#R1">=</a> <font color="Olive">s1</font> &amp; <font color="Maroon">1F</font> <a href="group_1.html#K1">*</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Olive">s1</font> &amp;  ex <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">uM</font> #) st <br/>( <font color="Olive">s1</font> <a href="group_1.html#K1">*</a> <font color="Olive">s2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  &amp; <font color="Olive">s2</font> <a href="group_1.html#K1">*</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">uM</font> #);<br/>

<a NAME="E1:45_1_1_3"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">s1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">1F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">1F</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">s1</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">k</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="E3:45_1_1_3_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
 <b>by </b><i/>;<br/>
<i><font color="Green">E69</font></i>: <font color="Maroon">e1</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> = 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 0</span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 0</span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i>, <a class="txt" href="uniroots.html#E4"><i><font color="Green">Lemma46</font></i></a>, </i>
<br/>.= 
<font color="Maroon">s1</font>
<b>by </b><i><a class="txt" href="uniroots.html#E4"><i><font color="Green">Lemma46</font></i></a>, </i>
;<br/>
<i><font color="Green">E70</font></i>: <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">e1</font> = 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 0</span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 0</span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i>, <a class="txt" href="uniroots.html#E4"><i><font color="Green">Lemma46</font></i></a>, </i>
<br/>.= 
<font color="Maroon">s1</font>
<b>by </b><i><a class="txt" href="uniroots.html#E4"><i><font color="Green">Lemma46</font></i></a>, </i>
;<br/>
<a NAME="E6:45_1_1_3_1"/>
 <a href="binop_2.html#K29">multcomplex</a>  <a href="funct_2.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<a NAME="E7:45_1_1_3_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">e1</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:45_1_1_3_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">1F</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:45_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">1F</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">1F</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E10:45_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">1F</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">s1</font>,<font color="Maroon">1F</font>
 
;<br/>
<a NAME="E11:45_1_1_3_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">s1</font>,<font color="Maroon">1F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:45_1_1_3_1"/>
 <a href="binop_2.html#K29">multcomplex</a>  <a href="funct_2.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<a NAME="E13:45_1_1_3_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span>,<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font>
 
<b>by </b><i><a class="ref" href="uniroots.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E14:45_1_1_3_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">1F</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E15:45_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">1F</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">1F</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E16:45_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">1F</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">1F</font>,<font color="Maroon">s1</font>
 
;<br/>
<a NAME="E17:45_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">1F</font>,<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E18:45_1_1_3_1"/>
( <font color="Maroon">s1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">1F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">1F</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:45_1_1_3"/>
 ex <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">uM</font> #) st <br/>( <font color="Maroon">s1</font> <a href="group_1.html#K1">*</a> <font color="Olive">s2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  &amp; <font color="Olive">s2</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:45_1_1_3_2"/>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</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="E3:45_1_1_3_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">e1</font> = <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  ;<br/>
<a NAME="E5:45_1_1_3_2"/>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <font color="Olive">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_3_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">r</font> = <font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>;<br/>
<a NAME="E1:45_1_1_3_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<a NAME="E2:45_1_1_3_2_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:45_1_1_3_2_1"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E5:45_1_1_3_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:45_1_1_3_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="uniroots.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:45_1_1_3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T63">NAT_1:63</a></i>;<br/>
<b>hence </b><a NAME="E8:45_1_1_3_2_1"/>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <font color="Olive">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">j</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="E7:45_1_1_3_2"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>
<b>set </b><font color="Maroon">ss2</font> = <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">s2</font> = <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">uM</font> #) <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">e2</font> = <font color="Maroon">s2</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  ;<br/>
<a NAME="E10:45_1_1_3_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">e1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:45_1_1_3_2"/>
 <a href="binop_2.html#K29">multcomplex</a>  <a href="funct_2.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<a NAME="E12:45_1_1_3_2"/><b>then </b><i><font color="Green">E170</font></i>: 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">e1</font>,<font color="Maroon">e2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i>, , <a class="ref" href="uniroots.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E13:45_1_1_3_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E14:45_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E15:45_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">s1</font>,<font color="Maroon">s2</font>
 
;<br/>
<a NAME="E16:45_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">s1</font>,<font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E4"><i><font color="Green">Lemma46</font></i></a>, </i>;<br/>
<a NAME="E17:45_1_1_3_2"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">s1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> 
 
;<br/>
<a NAME="E18:45_1_1_3_2"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">e2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E19:45_1_1_3_2"/>
 <a href="binop_2.html#K29">multcomplex</a>  <a href="funct_2.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<a NAME="E20:45_1_1_3_2"/><b>then </b><i><font color="Green">E188</font></i>: 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">e2</font>,<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E21:45_1_1_3_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s2</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:45_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s2</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s2</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E23:45_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s2</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">s2</font>,<font color="Maroon">s1</font>
 
;<br/>
<a NAME="E24:45_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">s2</font>,<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E4"><i><font color="Green">Lemma46</font></i></a>, </i>;<br/>
<a NAME="E25:45_1_1_3_2"/><b>then </b>
<font color="Maroon">s2</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> 
 
;<br/>
<b>hence </b><a NAME="E26:45_1_1_3_2"/>
 ex <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">uM</font> #) st <br/>( <font color="Maroon">s1</font> <a href="group_1.html#K1">*</a> <font color="Olive">s2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  &amp; <font color="Olive">s2</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_1_1_3"/>
( <font color="Maroon">s1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">1F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">1F</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp;  ex <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">uM</font> #) st <br/>( <font color="Maroon">s1</font> <a href="group_1.html#K1">*</a> <font color="Olive">s2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  &amp; <font color="Olive">s2</font> <a href="group_1.html#K1">*</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E15:45_1_1"/><b>then </b>
 <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">uM</font> #) is   <a href="group_1.html#NM1">Group</a>
 
<b>by </b><i><a class="ref" href="uniroots.html#T2" target="_self">Th2</a>, <a class="ref" href="group_1.html#T5">GROUP_1:5</a></i>;<br/>
<b>hence </b><a NAME="E16:45_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a> st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  &amp; the <a href="group_1.html#U1">mult</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a>  <a href="realset1.html#K1">||</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span> )
 ;<br/>


</div>
