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<b>let </b><font color="Maroon">L</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:67"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E2:67"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="polynom8.html#R1" target="_self">is_primitive_root_of_degree</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 ;<br/>

<b>assume </b><a NAME="E3:67"/><i><font color="Green">E39</font></i>: 
 <a href="polynom8.html#K1" target="_self">emb</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">L</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom8.html#K2" target="_self">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">q</font>,<font color="Maroon">x</font>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">L</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <span class="p2">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">x</font>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">L</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">L</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">L</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="polynom5.html#T28">POLYNOM5:28</a></i>
;<br/>
<b>hence </b><a NAME="E5:67"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">L</font></span>)</span> <a href="vectsp_1.html#K4">"</a> </span>)</span> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="polynom8.html#K2" target="_self">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">q</font>,<font color="Maroon">x</font>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font>
 ;<br/>


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