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<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <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">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <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">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  ;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:33_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:33_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:33_1_1"><i><font color="Green">E31</font></i></a>, </i>;<br/><a NAME="E6:33_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <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">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <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">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="uniroots.html#E2"><i><font color="Green">Lemma32</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:33_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <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">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <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">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span> 
 ;<br/><a NAME="E3:33_1"/><b>then </b>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 ;<br/><b>hence </b><a NAME="E4:33_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:33"/>
<font color="Maroon">n</font> <a href="uniroots.html#K2" target="_self">-roots_of_1</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <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">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <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">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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