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<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">x</font> be   <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> ;<br/>



<b>reconsider </b><font color="Maroon" title="c3">z</font> =  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  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>  ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c2">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> 
 ;<br/><div><i><font color="Green" title="E27">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 ;<br/><a NAME="E2:31_1_1"/><b>then </b>
<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="c2">x</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="complfld.html#T9" title="COMPLFLD:th.9">COMPLFLD:9</a>, <a class="ref" href="complfld.html#T10" title="COMPLFLD:th.10">COMPLFLD:10</a>, <a class="ref" href="vectsp_1.html#T95" title="VECTSP_1:th.95">VECTSP_1:95</a></i>;<br/><a NAME="E3:31_1_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> is  not    <a href="comptrig.html#M1" title="COMPTRIG:mode.1">CRoot</a> of <font color="Maroon" title="c1">n</font>, <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="complfld.html#D2" title="COMPLFLD:def.2">COMPLFLD:def 2</a></i>;<br/><b>hence </b><a NAME="E4:31_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E1:31_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="uniroots.html#T24" target="_self" title="UNIROOTS:th.24">Th24</a></i>;<br/></div><b>end;</b></div><a NAME="E3:31_1"/><b>then </b><i><font color="Green" title="E28">A3</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <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"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span> <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> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</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="complfld.html#T9" title="COMPLFLD:th.9">COMPLFLD:9</a>, <a class="ref" href="comptrig.html#D1" title="COMPTRIG:def.1">COMPTRIG:def 1</a></i>;<br/><a NAME="E4:31_1"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c2">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"><span class="p3"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></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#E2:31_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="complfld.html#T9" title="COMPLFLD:th.9">COMPLFLD:9</a>, <a class="ref" href="comptrig.html#D1" title="COMPTRIG:def.1">COMPTRIG:def 1</a></i>;<br/><a NAME="E5:31_1"/>
<font color="Maroon" title="c2">x</font> is    <a href="comptrig.html#M1" title="COMPTRIG:mode.1">CRoot</a> of <font color="Maroon" title="c1">n</font>, <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <b>by </b><i><a class="txt" href="uniroots.html#E1:31_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="uniroots.html#T24" target="_self" title="UNIROOTS:th.24">Th24</a></i>;<br/><a NAME="E6:31_1"/><b>then </b>
<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="c2">x</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="complfld.html#D2" title="COMPLFLD:def.2">COMPLFLD:def 2</a></i>;<br/><a NAME="E7:31_1"/><b>then </b><i><font color="Green" title="E30">A5</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E2:31_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="complfld.html#T97" title="COMPLFLD:th.97">COMPLFLD:97</a>, <a class="ref" href="polynom5.html#T8" title="POLYNOM5:th.8">POLYNOM5:8</a></i>;<br/><div><i><font color="Green" title="E31">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_1_2"/><i><font color="Green" title="E32">A7</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:31_1_2"/><i><font color="Green" title="E33">A8</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="uniroots.html#E2:31_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="complfld.html#T96" title="COMPLFLD:th.96">COMPLFLD:96</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_2_1"/>
( <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 or <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="uniroots.html#E1:31_1_2"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1_1"><b>suppose </b></a><a NAME="E1:31_1_2_1_1"/><i><font color="Green" title="E34">A9</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c4">n'</font> = <font color="Maroon" title="c1">n</font> as   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a> ;<br/><a NAME="E3:31_1_2_1_1"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="prepower.html#K7" title="PREPOWER:func.7">#Q</a> <font color="Maroon" title="c4">n'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E2:31_1_2"><i><font color="Green" title="E33">A8</font></i></a>, <a class="txt" href="uniroots.html#E1:31_1_2_1_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="prepower.html#T76" title="PREPOWER:th.76">PREPOWER:76</a></i>;<br/><a NAME="E4:31_1_2_1_1"/><b>then </b>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E2:31_1_2"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="prepower.html#T60" title="PREPOWER:th.60">PREPOWER:60</a></i>;<br/><b>hence </b><a NAME="E5:31_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E7:31_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="uniroots.html#E2:31_1_2"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="power.html#T46" title="POWER:th.46">POWER:46</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1_2"><b>suppose </b></a><a NAME="E1:31_1_2_1_2"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="uniroots.html#E7:31_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="power.html#T40" title="POWER:th.40">POWER:40</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c4">z</font> = <font color="Maroon" title="c2">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></i>;<br/><a NAME="E10:31_1"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E11:31_1"/><b>then </b><i><font color="Green" title="E32">A10</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="power.html#T47" title="POWER:th.47">POWER:47</a></i>;<br/><a NAME="E12:31_1"/><i><font color="Green" title="E33">A11</font></i>: 
<font color="Maroon" title="c4">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <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"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c4">z</font></span>)</span></span>)</span></span>)</span></span>)</span> <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="comptrig.html#T72" title="COMPTRIG:th.72">COMPTRIG:72</a></i>;<br/><a NAME="E13:31_1"/>
<font color="Maroon" title="c2">x</font> is    <a href="comptrig.html#M1" title="COMPTRIG:mode.1">CRoot</a> of <font color="Maroon" title="c1">n</font>, <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <b>by </b><i><a class="txt" href="uniroots.html#E1:31_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="uniroots.html#T24" target="_self" title="UNIROOTS:th.24">Th24</a></i>;<br/><a NAME="E14:31_1"/><b>then </b>
<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="c2">x</font>,<font color="Maroon" title="c1">n</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">1,0</span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
 <b>by </b><i><a class="ref" href="complfld.html#T10" title="COMPLFLD:th.10">COMPLFLD:10</a>, <a class="ref" href="complfld.html#D2" title="COMPLFLD:def.2">COMPLFLD:def 2</a></i>;<br/><a NAME="E15:31_1"/><b>then </b><i><font color="Green" title="E34">A12</font></i>: 
<font color="Maroon" title="c4">z</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">n</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">1,0</span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
 <b>by </b><i><a class="ref" href="complfld.html#T112" title="COMPLFLD:th.112">COMPLFLD:112</a></i>;<br/><a NAME="E16:31_1"/><b>then </b><i><font color="Green" title="E35">A13</font></i>: 
(  <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="uniroots.html#E7:31_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="uniroots.html#E11:31_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="uniroots.html#E12:31_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="complex1.html#T163" title="COMPLEX1:th.163">COMPLEX1:163</a></i>;<br/><b>set </b><font color="Maroon" title="c5">d</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <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></span>)</span>;<br/><b>set </b><font color="Maroon" title="c6">m</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4">(<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></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>;<br/><b>consider </b><font color="Maroon" title="c7">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E18:31_1"/><i><font color="Green" title="E36">A14</font></i>: 
( <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p0">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <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></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c7">r</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="complex2.html#T2" title="COMPLEX2:th.2">COMPLEX2:2</a>, <a class="ref" href="comptrig.html#T21" title="COMPTRIG:th.21">COMPTRIG:21</a></i>;<br/><b>set </b><font color="Maroon" title="c8">a</font> = <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"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <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></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>;<br/><a NAME="E19:31_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:31_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="real_2.html#T121" title="REAL_2:th.121">REAL_2:121</a></i>;<br/><a NAME="E20:31_1"/><b>then </b><i><font color="Green" title="E37">A15</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="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</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>
 <b>by </b><i><a class="ref" href="comptrig.html#T21" title="COMPTRIG:th.21">COMPTRIG:21</a>, <a class="ref" href="real_2.html#T125" title="REAL_2:th.125">REAL_2:125</a></i>;<br/><b>set </b><font color="Maroon" title="c9">m2</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <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></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>;<br/><a NAME="E21:31_1"/><i><font color="Green" title="E38">A16</font></i>: 
0 <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <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"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <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></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="uniroots.html#E20:31_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E22:31_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <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></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <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></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><a NAME="E23:31_1"/><b>then </b>
not <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <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></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1
 <b>by </b><i><a class="txt" href="uniroots.html#E21:31_1"><i><font color="Green" title="E38">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E24:31_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <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></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">m</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <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></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> 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/><i><font color="Green" title="E39">A17</font></i>:  <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> = 
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">r</font></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E18:31_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="sin_cos.html#D23" title="SIN_COS:def.23">SIN_COS:def 23</a></i>
<br/>.= 
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r</font>
<b>by </b><i><a class="ref" href="sin_cos2.html#T11" title="SIN_COS2:th.11">SIN_COS2:11</a></i>
<br/>.= 
 <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> <font color="Maroon" title="c7">r</font>
<b>by </b><i><a class="ref" href="sin_cos.html#D23" title="SIN_COS:def.23">SIN_COS:def 23</a></i>
;<br/> <a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> = 
<a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">r</font></span>)</span>
<b>by </b><i><a class="txt" href="uniroots.html#E18:31_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="sin_cos.html#D21" title="SIN_COS:def.21">SIN_COS:def 21</a></i>
<br/>.= 
<a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">r</font>
<b>by </b><i><a class="ref" href="sin_cos2.html#T10" title="SIN_COS2:th.10">SIN_COS2:10</a></i>
<br/>.= 
 <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> <font color="Maroon" title="c7">r</font>
<b>by </b><i><a class="ref" href="sin_cos.html#D21" title="SIN_COS:def.21">SIN_COS:def 21</a></i>
;<br/><a NAME="E28:31_1"/><b>then </b>
( <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  )
 <b>by </b><i><a class="txt" href="uniroots.html#E16:31_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="txt" href="uniroots.html#E18:31_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="comptrig.html#T33" title="COMPTRIG:th.33">COMPTRIG:33</a></i>;<br/><a NAME="E29:31_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</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#K24" title="BINOP_2:func.24">*</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c10">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="txt" href="uniroots.html#E7:31_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="uniroots.html#E11:31_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="uniroots.html#E12:31_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="uniroots.html#E15:31_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="uniroots.html#E18:31_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="uniroots.html#E26:31_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="complex1.html#T163" title="COMPLEX1:th.163">COMPLEX1:163</a>, <a class="ref" href="sin_cos.html#T82" title="SIN_COS:th.82">SIN_COS:82</a></i>;<br/><a NAME="E30:31_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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="c1">n</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c10">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#T84" title="XCMPLX_1:th.84">XCMPLX_1:84</a></i>;<br/><a NAME="E31:31_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c10">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#T89" title="XCMPLX_1:th.89">XCMPLX_1:89</a></i>;<br/><b>hence </b><a NAME="E32:31_1"/>
 ex <font color="Olive" title="b1">k</font> being   <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>  st <font color="Maroon" title="c2">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="Olive" title="b1">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="Olive" title="b1">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>
 <b>by </b><i><a class="txt" href="uniroots.html#E4:31_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="uniroots.html#E8:31_1"><i><font color="Green" title="E31">A6</font></i></a></i>;<br/></div>
<b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c4">k</font> being    <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> such that </b><a NAME="E2:31_2"/><i><font color="Green" title="E26">A18</font></i>: 
<font color="Maroon" title="c2">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="c4">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="c4">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/><b>set </b><font color="Maroon" title="c5">1F</font> =  <a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>;<br/><a NAME="E3:31_2"/>
0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:31_2"/><b>then </b><i><font color="Green" title="E27">A19</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="power.html#K2" title="POWER:func.2">-root</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="power.html#T7" title="POWER:th.7">POWER:7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">z</font> =  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></i>;<br/><a NAME="E6:31_2"/>
<font color="Maroon" title="c2">x</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="c1">n</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <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"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p0">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <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="c4">k</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>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="power.html#K2" title="POWER:func.2">-root</a> <span class="p4"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="comptrig.html#K1" title="COMPTRIG:func.1">Arg</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p0">(<span class="default"><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="c4">k</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>)</span> <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="txt" href="uniroots.html#E2:31_2"><i><font color="Green" title="E26">A18</font></i></a>, <a class="txt" href="uniroots.html#E4:31_2"><i><font color="Green" title="E27">A19</font></i></a>, <a class="ref" href="complfld.html#T10" title="COMPLFLD:th.10">COMPLFLD:10</a>, <a class="ref" href="complfld.html#T97" title="COMPLFLD:th.97">COMPLFLD:97</a>, <a class="ref" href="comptrig.html#T57" title="COMPTRIG:th.57">COMPTRIG:57</a></i>;<br/><a NAME="E7:31_2"/><b>then </b>
<font color="Maroon" title="c2">x</font> is    <a href="comptrig.html#M1" title="COMPTRIG:mode.1">CRoot</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="ref" href="comptrig.html#T75" title="COMPTRIG:th.75">COMPTRIG:75</a></i>;<br/><b>hence </b><a NAME="E8:31_2"/>
<font color="Maroon" title="c2">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> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:31"/>
(  ex <font color="Olive" title="b1">k</font> being   <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>  st <font color="Maroon" title="c2">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="Olive" title="b1">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="Olive" title="b1">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> implies <font color="Maroon" title="c2">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>  )
 ;<br/>


</div>
