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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<i><font color="Green">E31</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">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K9">+</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">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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">i</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font>

;<br/>
<a NAME="E2:14"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<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">i</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><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">i</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K24">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">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T80">SIN_COS:80</a></i>;<br/>
<a NAME="E3:14"/>
<span class="p1">(<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">i</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <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">i</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="sin_cos.html#K21">sin</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">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="sin_cos.html#T80">SIN_COS:80</a></i>;<br/>
<b>then </b><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">i</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">i</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> <a href="group_1.html#K10">*</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">j</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> = 
<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">i</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</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">i</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</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="ref" href="hahnban1.html#T11">HAHNBAN1:11</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E5:14"/>
<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">i</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">i</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> <a href="group_1.html#K10">*</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">j</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K35">PI</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></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>
 ;<br/>


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