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

<b>set </b><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">E39</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">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">E40</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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#E1:45_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">E41</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">c<sub>1</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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#E5:45_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E8:45_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></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">E43</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:45_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">E44</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  )
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:45_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="uniroots.html#E3:45_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></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">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="txt" href="uniroots.html#E4:45_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<a NAME="E7:45_1_1_1"/><i><font color="Green">E45</font></i>: 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K5">*</a> <font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:45_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="uniroots.html#E1:45_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="uniroots.html#E3:45_1_1_1"><i><font color="Green">E43</font></i></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">c<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1_1"><i><font color="Green">E45</font></i></a></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">c<sub>1</sub></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">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 <b>by </b><i><a class="txt" href="uniroots.html#E3:45_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="uniroots.html#E4:45_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="uniroots.html#T28" target="_self">Th28</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  <b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>
<a NAME="E10:45_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>5</sub></font> =  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font> #);<br/>
<a NAME="E11:45_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font> #) holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>3</sub></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">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font> #);<br/>

<a NAME="E1:45_1_1_2"/>
( <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></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">c<sub>8</sub></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">E44</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  &amp; <font color="Maroon">c<sub>7</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  &amp; <font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></font> =  <a href="binop_2.html#K29">multcomplex</a> ;<br/>
<a NAME="E3:45_1_1_2"/><i><font color="Green">E45</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E41</font></i></a>, <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">E46</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">c<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:45_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">c<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:45_1_1_2"><i><font color="Green">E45</font></i></a>, <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">c<sub>6</sub></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">c<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="uniroots.html#E10:45_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then </b><i><font color="Green">E47</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">c<sub>1</sub></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">c<sub>6</sub></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">c<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<a href="binop_2.html#K29">multcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="uniroots.html#E1:45_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 href="binop_2.html#K29">multcomplex</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<span class="p2">(<span class="default"><a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="uniroots.html#E4:45_1_1_2"><i><font color="Green">E46</font></i></a></i>
<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>

;<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">c<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:45_1_1_2"><i><font color="Green">E45</font></i></a>, <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">c<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="uniroots.html#E10:45_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<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">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> = 
<a href="binop_2.html#K29">multcomplex</a>  <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">c<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="uniroots.html#E1:45_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 href="binop_2.html#K29">multcomplex</a>  <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"><a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="uniroots.html#E4:45_1_1_2"><i><font color="Green">E46</font></i></a></i>
<br/>.= 
<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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>

;<br/>
<b>hence </b><a NAME="E11:45_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E2:45_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="uniroots.html#E7:45_1_1_2"><i><font color="Green">E47</font></i></a>, <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">c<sub>6</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font> #) <b>by </b><i><a class="ref" href="uniroots.html#T25" target="_self">Th25</a></i>;<br/>
<i><font color="Green">E44</font></i>:  <a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a>  = 
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>
<br/>.= 
<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#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> 0</span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></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#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> 0</span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="ref" href="sin_cos.html#T34">SIN_COS:34</a></i>
;<br/>
<a NAME="E14:45_1_1"/>
for <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font> #) holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font> #) st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></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">b<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></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_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font> #);<br/>

<a NAME="E1:45_1_1_4"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_4_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></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">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:45_1_1_4_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>8</sub></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#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></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#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
 <b>by </b><i><a class="ref" href="uniroots.html#T27" target="_self">Th27</a></i>;<br/>
<i><font color="Green">E47</font></i>: <font color="Maroon">c<sub>8</sub></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#K33">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">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 0</span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></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#K33">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">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 0</span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="txt" href="uniroots.html#E13:45_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="uniroots.html#E3:45_1_1_4_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="uniroots.html#T13" target="_self">Th13</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="uniroots.html#E3:45_1_1_4_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="uniroots.html#T12" target="_self">Th12</a></i>
;<br/>
<i><font color="Green">E48</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">c<sub>8</sub></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#K33">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">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 0</span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></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#K33">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">c<sub>9</sub></font> <a href="binop_2.html#K23">+</a> 0</span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i><a class="txt" href="uniroots.html#E13:45_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="uniroots.html#E3:45_1_1_4_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="uniroots.html#T13" target="_self">Th13</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="uniroots.html#E3:45_1_1_4_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="uniroots.html#T12" target="_self">Th12</a></i>
;<br/>
<a NAME="E6:45_1_1_4_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_4_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E4:45_1_1_4_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E8:45_1_1_4_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:45_1_1_4_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">c<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:45_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_4_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">c<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<a NAME="E11:45_1_1_4_1"/><b>then </b><i><font color="Green">E50</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">c<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1_4_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E12:45_1_1_4_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_4_1"/><b>then </b><i><font color="Green">E51</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">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E5:45_1_1_4_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E14:45_1_1_4_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E15:45_1_1_4_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">c<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:45_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_4_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">c<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
;<br/>
<a NAME="E17:45_1_1_4_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">c<sub>1</sub></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">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E13:45_1_1_4_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:45_1_1_4_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E11:45_1_1_4_1"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:45_1_1_4"/>
ex <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font> #) st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></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_4_2"><b>proof </b></a><div class="add">

<a NAME="E1:45_1_1_4_2"/>
<font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:45_1_1_4_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>7</sub></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#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></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#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
 <b>by </b><i><a class="ref" href="uniroots.html#T27" target="_self">Th27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></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#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></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#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></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_4_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_4_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:45_1_1_4_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></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_4_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></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">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:45_1_1_4_2_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></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_4_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></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">c<sub>8</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:45_1_1_4_2_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="uniroots.html#E4:45_1_1_4_2_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E6:45_1_1_4_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></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="E7:45_1_1_4_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:45_1_1_4_2"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></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#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></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#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></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#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></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#K33">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font> #) <b>by </b><i><a class="ref" href="uniroots.html#T27" target="_self">Th27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></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_4_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>13</sub></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#K33">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">c<sub>10</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></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#K33">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">c<sub>10</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
 
<b>by </b><i><a class="ref" href="uniroots.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E11:45_1_1_4_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_4_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>13</sub></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#E13:45_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="uniroots.html#E7:45_1_1_4_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="uniroots.html#E10:45_1_1_4_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E13:45_1_1_4_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E14:45_1_1_4_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">c<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:45_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_4_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">c<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 
;<br/>
<a NAME="E16:45_1_1_4_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">c<sub>1</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></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#E3:45_1_1_4_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="uniroots.html#E12:45_1_1_4_2"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E17:45_1_1_4_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></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_4_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>9</sub></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#K33">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">c<sub>10</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></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#K33">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">c<sub>10</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
 
<b>by </b><i><a class="ref" href="uniroots.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E19:45_1_1_4_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_4_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<a href="binop_2.html#K29">multcomplex</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>9</sub></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#E13:45_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="uniroots.html#E7:45_1_1_4_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="uniroots.html#E18:45_1_1_4_2"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E21:45_1_1_4_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E7:45_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:45_1_1_4_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">c<sub>1</sub></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">c<sub>12</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>12</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:45_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_4_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">c<sub>1</sub></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">c<sub>12</sub></font>,<font color="Maroon">c<sub>7</sub></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">c<sub>12</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
;<br/>
<a NAME="E24:45_1_1_4_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">c<sub>1</sub></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">c<sub>12</sub></font>,<font color="Maroon">c<sub>7</sub></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#E3:45_1_1_4_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="uniroots.html#E20:45_1_1_4_2"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E25:45_1_1_4_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></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_4_2"/>
ex <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font> #) st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></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#E17:45_1_1_4_2"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_1_1_4"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font> #) st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>7</sub></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#E1:45_1_1_4"><i><font color="Green">E45</font></i></a></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font> #) is   <a href="group_1.html#NM1">Group</a>
 
<b>by </b><i><a class="txt" href="uniroots.html#E11:45_1_1"><i><font color="Green">E43</font></i></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">c<sub>1</sub></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">c<sub>1</sub></font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> </span>)</span> )
 ;<br/>


</div>
