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<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>;<br/>





<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">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" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">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" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
 ;<br/>

<a NAME="E2:39"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T32" title="INT_2:th.32">INT_2:32</a></i>;<br/>
<a NAME="E3:39"/>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E4:39"/><b>then </b><i><font color="Green" title="E35">A3</font></i>: 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T71" title="NEWTON:th.71">NEWTON:71</a></i>;<br/>
<a NAME="E5:39"/><b>then </b><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">kgn</font> = <font color="Maroon" title="c2">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="uniroots.html#E4:39"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></i>;<br/>
<a NAME="E7:39"/><i><font color="Green" title="E37">A5</font></i>: 
<font color="Maroon" title="c4">kgn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:39"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">vn</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:39"/><i><font color="Green" title="E38">A6</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">kgn</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">vn</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E2:39"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">vn</font> = <font color="Maroon" title="c5">vn</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E11:39"/>
<font color="Maroon" title="c2">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">k</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T31" title="INT_2:th.31">INT_2:31</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E13:39"/><i><font color="Green" title="E39">A7</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">kgn</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">i</font> = <font color="Maroon" title="c7">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<div><i><font color="Green" title="E40">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">kgn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:39_1"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">kgn</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 0</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">kgn</font></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E5:39"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/><b>hence </b><a NAME="E3:39_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E5:39"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="uniroots.html#E7:39"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:39"/><i><font color="Green" title="E41">A9</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">kgn</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c6">vn</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 0
 
<b>by </b><i><a class="txt" href="uniroots.html#E9:39"><i><font color="Green" title="E38">A6</font></i></a></i>;<br/>
<a NAME="E17:39"/><b>then </b><i><font color="Green" title="E42">A10</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">kgn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">vn</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E5:39"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="nat_d.html#D1" title="NAT_D:def.1">NAT_D:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  <b>by </b><i><a class="ref" href="uniroots.html#T22" target="_self" title="UNIROOTS:th.22">Th22</a></i>;<br/>
<i><font color="Green" title="E43">A11</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c6">vn</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">kgn</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c6">vn</font></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E13:39"><i><font color="Green" title="E39">A7</font></i></a>, <a class="ref" href="xcmplx_1.html#T83" title="XCMPLX_1:th.83">XCMPLX_1:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">kgn</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c6">vn</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="uniroots.html#E9:39"><i><font color="Green" title="E38">A6</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 0
<b>by </b><i><a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>
;<br/>
<div><i><font color="Green" title="E44">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_3"/>
<font color="Maroon" title="c6">vn</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><b>hence </b><a NAME="E2:39_3"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E9:39"><i><font color="Green" title="E38">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<i><font color="Green" title="E45">A13</font></i>: <font color="Maroon" title="c3">x</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">kgn</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">vn</font>
<b>by </b><i><a class="txt" href="uniroots.html#E17:39"><i><font color="Green" title="E42">A10</font></i></a>, <a class="ref" href="uniroots.html#T32" target="_self" title="UNIROOTS:th.32">Th32</a></i>
<br/>.= 
<font color="Maroon" title="c9">y</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">vn</font>
<b>by </b><i><a class="txt" href="uniroots.html#E20:39"><i><font color="Green" title="E44">A12</font></i></a>, <a class="ref" href="complfld.html#T112" title="COMPLFLD:th.112">COMPLFLD:112</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c6">vn</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c6">vn</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
<b>by </b><i><a class="txt" href="uniroots.html#E1:39"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="comptrig.html#T71" title="COMPTRIG:th.71">COMPTRIG:71</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">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" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 0</span>)</span></span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
<b>by </b><i><a class="txt" href="uniroots.html#E19:39"><i><font color="Green" title="E43">A11</font></i></a>, <a class="ref" href="complex2.html#T10" title="COMPLEX2:th.10">COMPLEX2:10</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T9" title="COMPLEX2:th.9">COMPLEX2:9</a>, <a class="ref" href="sin_cos.html#T34" title="SIN_COS:th.34">SIN_COS:34</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>
<b>by </b><i><a class="ref" href="uniroots.html#T20" target="_self" title="UNIROOTS:th.20">Th20</a>, <a class="ref" href="complfld.html#T10" title="COMPLFLD:th.10">COMPLFLD:10</a></i>
;<br/>
<a NAME="E22:39"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="uniroots.html#K2" title="UNIROOTS:func.2">-roots_of_1</a> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:39"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="uniroots.html#T27" target="_self" title="UNIROOTS:th.27">Th27</a></i>;<br/>
<a NAME="E23:39"/><b>then </b><i><font color="Green" title="E46">A14</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="group_1.html#NR1" title="GROUP_1:NR.1">is_not_of_order_0</a> 
 
<b>by </b><i><a class="ref" href="uniroots.html#T33" target="_self" title="UNIROOTS:th.33">Th33</a></i>;<br/>
<a NAME="E24:39"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c3">x</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 holds <br/><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">kgn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:39_5"/><i><font color="Green" title="E47">A15</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>
 <b>and </b><br/><a NAME="E2:39_5"/><i><font color="Green" title="E48">A16</font></i>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E3:39_5"/><i><font color="Green" title="E49">A17</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">m</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:39_5"><i><font color="Green" title="E48">A16</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">m</font> = <font color="Maroon" title="c10">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_5_1"/><i><font color="Green" title="E50">A18</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">vn</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:39_5_1_1"/><b>then </b><i><font color="Green" title="E51">A19</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E4:39_5_1_1"/><i><font color="Green" title="E52">A20</font></i>: 
( <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">kgn</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">kgn</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">b</font> )
 <b>and </b><br/><a NAME="E5:39_5_1_1"/><i><font color="Green" title="E53">A21</font></i>: 
<font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#T38" title="INT_2:th.38">INT_2:38</a></i>;<br/><a NAME="E6:39_5_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_5_1_1_1"/>
0 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">b</font>
 ;<br/>

<a NAME="E2:39_5_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">kgn</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">b</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E5:39"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E3:39_5_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E4:39_5_1_1"><i><font color="Green" title="E52">A20</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c14">bi</font> = <font color="Maroon" title="c13">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E8:39_5_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_5_1_1_2"/>
0 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">a</font>
 ;<br/>

<a NAME="E2:39_5_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">kgn</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">a</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E5:39"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E3:39_5_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E4:39_5_1_1"><i><font color="Green" title="E52">A20</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c15">ai</font> = <font color="Maroon" title="c12">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E10:39_5_1_1"/><i><font color="Green" title="E54">A22</font></i>: 
<font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">bi</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E5:39"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="uniroots.html#E17:39"><i><font color="Green" title="E42">A10</font></i></a>, <a class="txt" href="uniroots.html#E1:39_5_1"><i><font color="Green" title="E50">A18</font></i></a>, <a class="txt" href="uniroots.html#E4:39_5_1_1"><i><font color="Green" title="E52">A20</font></i></a>, <a class="ref" href="nat_d.html#T18" title="NAT_D:th.18">NAT_D:18</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E12:39_5_1_1"/><i><font color="Green" title="E55">A23</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E2:39_5_1_1"><i><font color="Green" title="E51">A19</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E13:39_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">kgn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">kgn</font></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:39_5_1_1"><i><font color="Green" title="E52">A20</font></i></a>, <a class="txt" href="uniroots.html#E12:39_5_1_1"><i><font color="Green" title="E55">A23</font></i></a></i>;<br/><a NAME="E14:39_5_1_1"/><b>then </b>
<font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">kgn</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">kgn</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E5:39"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/><a NAME="E15:39_5_1_1"/><b>then </b>
<font color="Maroon" title="c11">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">j</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">b</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E5:39"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/><a NAME="E16:39_5_1_1"/><b>then </b>
<font color="Maroon" title="c14">bi</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c11">m</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c15">ai</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E17:39_5_1_1"/><b>then </b>
<font color="Maroon" title="c13">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E5:39_5_1_1"><i><font color="Green" title="E53">A21</font></i></a>, <a class="ref" href="int_2.html#T40" title="INT_2:th.40">INT_2:40</a></i>;<br/><b>hence </b><a NAME="E18:39_5_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E3:39_5"><i><font color="Green" title="E49">A17</font></i></a>, <a class="txt" href="uniroots.html#E10:39_5_1_1"><i><font color="Green" title="E54">A22</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/></div><b>end;</b></div><a NAME="E3:39_5_1"/><b>then </b><i><font color="Green" title="E51">A24</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/><i><font color="Green" title="E52">A25</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c11">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c11">m</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T83" title="XCMPLX_1:th.83">XCMPLX_1:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T78" title="XCMPLX_1:th.78">XCMPLX_1:78</a></i>
;<br/><div><i><font color="Green" title="E53">A26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_5_1_3"/>
<font color="Maroon" title="c11">m</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><b>hence </b><a NAME="E2:39_5_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E2:39_5"><i><font color="Green" title="E48">A16</font></i></a></i>;<br/></div><b>end;</b></div><i><font color="Green" title="E54">A27</font></i>: <font color="Maroon" title="c3">x</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c11">m</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c11">m</font>
<b>by </b><i><a class="ref" href="uniroots.html#T32" target="_self" title="UNIROOTS:th.32">Th32</a></i>
<br/>.= 
<font color="Maroon" title="c9">y</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c11">m</font>
<b>by </b><i><a class="txt" href="uniroots.html#E5:39_5_1"><i><font color="Green" title="E53">A26</font></i></a>, <a class="ref" href="complfld.html#T112" title="COMPLFLD:th.112">COMPLFLD:112</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">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" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
<b>by </b><i><a class="txt" href="uniroots.html#E1:39"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="uniroots.html#E4:39_5_1"><i><font color="Green" title="E52">A25</font></i></a>, <a class="ref" href="comptrig.html#T71" title="COMPTRIG:th.71">COMPTRIG:71</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">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" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">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" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
<b>by </b><i><a class="ref" href="uniroots.html#T12" target="_self" title="UNIROOTS:th.12">Th12</a></i>
;<br/><a NAME="E7:39_5_1"/>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="uniroots.html#K1" title="UNIROOTS:func.1">MultGroup</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default">1,0</span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
 <b>by </b><i><a class="ref" href="uniroots.html#T20" target="_self" title="UNIROOTS:th.20">Th20</a>, <a class="ref" href="complfld.html#T10" title="COMPLFLD:th.10">COMPLFLD:10</a></i>;<br/><a NAME="E8:39_5_1"/><b>then </b><i><font color="Green" title="E55">A28</font></i>: 
 <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E1:39_5"><i><font color="Green" title="E47">A15</font></i></a>, <a class="txt" href="uniroots.html#E6:39_5_1"><i><font color="Green" title="E54">A27</font></i></a>, <a class="ref" href="complex1.html#T163" title="COMPLEX1:th.163">COMPLEX1:163</a></i>;<br/><a NAME="E9:39_5_1"/>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:39_5_1"><i><font color="Green" title="E51">A24</font></i></a>, <a class="ref" href="comptrig.html#T21" title="COMPTRIG:th.21">COMPTRIG:21</a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E10:39_5_1"/><b>then </b><i><font color="Green" title="E56">A29</font></i>: 
0 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/><a NAME="E11:39_5_1"/>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:39_5_1"><i><font color="Green" title="E51">A24</font></i></a>, <a class="ref" href="comptrig.html#T21" title="COMPTRIG:th.21">COMPTRIG:21</a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E12:39_5_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="real_1.html#T73" title="REAL_1:th.73">REAL_1:73</a></i>;<br/><a NAME="E13:39_5_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/><b>hence </b><a NAME="E14:39_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E8:39_5_1"><i><font color="Green" title="E55">A28</font></i></a>, <a class="txt" href="uniroots.html#E10:39_5_1"><i><font color="Green" title="E56">A29</font></i></a>, <a class="ref" href="comptrig.html#T79" title="COMPTRIG:th.79">COMPTRIG:79</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:39_5"/>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">kgn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E5:39"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="uniroots.html#E16:39"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="nat_d.html#D1" title="NAT_D:def.1">NAT_D:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E25:39"/>
 <a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E15:39"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="uniroots.html#E21:39"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="uniroots.html#E23:39"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="group_1.html#D12" title="GROUP_1:def.12">GROUP_1:def 12</a></i>;<br/>


</div>
