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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>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> , <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:32"/><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> 
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">y</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>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>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">x</font> <a href="binop_2.html#K5">*</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<a NAME="E6:32"/>
 <a href="binop_2.html#K29">multcomplex</a>  <a href="funct_2.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<a NAME="E7:32"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="binop_2.html#D5">BINOP_2:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">i</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="E9:32"/><i><font color="Green">E40</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">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>
 <b>by </b><i><a class="ref" href="uniroots.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>consider </b><font color="Maroon">j</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="E11:32"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">b</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">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>
 <b>by </b><i>, </i>;<br/>
<a NAME="E12:32"/>
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</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"><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><a class="txt" href="uniroots.html#E2"><i><font color="Green">Lemma32</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E13:32"/>
<font color="Maroon">x</font> <a href="binop_2.html#K5">*</a> <font color="Maroon">y</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><a class="txt" href="uniroots.html#E1:32"><i><font color="Green">E31</font></i></a>, </i>;<br/>


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