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<div class="add">

<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="E1:66"/><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="E2:66"/><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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1"/>
(  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0 or  <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0 or (  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1"><b>suppose </b></a><a NAME="E1:66_1_1"/><i><font color="Green">E39</font></i>: 
(  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0 or  <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_1_1"/>
(  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0 or  <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_1_1_1"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:66_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="polynom4.html#T8">POLYNOM4:8</a></i>;<br/><a NAME="E3:66_1_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="polynom3.html#T35">POLYNOM3:35</a></i>;<br/><a NAME="E4:66_1_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p2">(<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="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="polynom5.html#T29">POLYNOM5:29</a></i>;<br/><a NAME="E5:66_1_1_1_1"/>
 <a href="polynom8.html#K6" target="_self">DFT</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</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>,<font color="Maroon">x</font>,<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"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="polynom8.html#K6" target="_self">DFT</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:66_1_1_1_1"/>
 <a href="polynom8.html#K6" target="_self">DFT</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</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>,<font color="Maroon">x</font>,<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"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p2">(<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="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>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_1_1_2"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:66_1_1_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="polynom4.html#T8">POLYNOM4:8</a></i>;<br/><a NAME="E3:66_1_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="polynom3.html#T35">POLYNOM3:35</a></i>;<br/><a NAME="E4:66_1_1_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p2">(<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="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="polynom5.html#T29">POLYNOM5:29</a></i>;<br/><a NAME="E5:66_1_1_1_2"/>
 <a href="polynom8.html#K6" target="_self">DFT</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</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>,<font color="Maroon">x</font>,<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"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="polynom8.html#K6" target="_self">DFT</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:66_1_1_1_2"/>
 <a href="polynom8.html#K6" target="_self">DFT</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</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>,<font color="Maroon">x</font>,<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"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p2">(<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="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>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2"><b>suppose </b></a><a NAME="E1:66_1_2"/>
(  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:66_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
(  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><a NAME="E3:66_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="polynom8.html#E6:66_1_2"><i><font color="Green">E62</font></i></a></i>;<br/><a NAME="E4:66_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 ;<br/><b>set </b><font color="Maroon">v1</font> =  <a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>;<br/><b>set </b><font color="Maroon">v2</font> =  <a href="polynom8.html#NK8" target="_self">VM</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>;<br/><b>set </b><font color="Maroon">v</font> =  <a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>;<br/><i><font color="Green">E61</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <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> = 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<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><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/><i><font color="Green">E62</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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 class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> = 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="polynom8.html#E9:66_1_2"><i><font color="Green">E66</font></i></a>, </i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<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><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/><i><font color="Green">E63</font></i>: <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <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="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<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"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<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="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <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><a class="txt" href="polynom8.html#E7:66_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="polynom8.html#E9:66_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="polynom8.html#E3:66_1_2"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p2">(<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="matrix_3.html#K6">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<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><a class="txt" href="polynom8.html#E7:66_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="polynom8.html#E9:66_1_2"><i><font color="Green">E66</font></i></a>, </i>
;<br/><a NAME="E8:66_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</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>,<font color="Maroon">x</font>,<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="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:66_1_2_4"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</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>,<font color="Maroon">x</font>,<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="normsp_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:66_1_2_4"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="algseq_1.html#R1">is_at_least_length_of</a>  <a href="polynom8.html#K6" target="_self">DFT</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>,<font color="Maroon">x</font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E15:66_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="algseq_1.html#D3">ALGSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:66_1_2_4"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</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>,<font color="Maroon">x</font>,<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="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="algseq_1.html#D4">ALGSEQ_1:def 4</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E66</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <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> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<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><a class="txt" href="polynom8.html#E11:66_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/><i><font color="Green">E67</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <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> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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 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><a class="txt" href="polynom8.html#E11:66_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="polynom8.html#E9:66_1_2"><i><font color="Green">E66</font></i></a>, </i>
;<br/><a NAME="E11:66_1_2"/><i><font color="Green">E68</font></i>: 
 <a href="algseq_1.html#K3">len</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> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E8:66_1_2"><i><font color="Green">E64</font></i></a>, </i>;<br/><a NAME="E12:66_1_2"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E6:66_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E13:66_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="algseq_1.html#K3">len</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> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E14:66_1_2"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <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> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</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>,<font color="Maroon">x</font>,<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_1.html#K1">.</a> <font color="Olive">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:66_1_2_7"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E2:66_1_2_7"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <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> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</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>,<font color="Maroon">x</font>,<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_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E15:66_1_2"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<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="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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 class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> is   <a href="matrix_1.html#NM6">Matrix</a> of 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>,1,<font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E16:66_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="polynom8.html#E15:66_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="polynom8.html#E9:66_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/><i><font color="Green">E72</font></i>:  <a href="polynom8.html#K5" target="_self">aConv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <span class="p3">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <span class="p4">(<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="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> = 
 <a href="polynom8.html#K5" target="_self">aConv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <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="matrix_3.html#K4">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span>,<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>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E9:66_1_2"><i><font color="Green">E66</font></i></a>, , , </i>
<br/>.= 
 <a href="polynom8.html#K5" target="_self">aConv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<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="matrix_3.html#K4">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <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> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E11:66_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="polynom8.html#E10:66_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="matrix_3.html#T35">MATRIX_3:35</a></i>
<br/>.= 
 <a href="polynom8.html#K5" target="_self">aConv</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="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</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="matrix_3.html#K4">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span>,<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>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E13:66_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="polynom8.html#E9:66_1_2"><i><font color="Green">E66</font></i></a>, </i>
<br/>.= 
 <a href="polynom8.html#K5" target="_self">aConv</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="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E6:66_1_2"><i><font color="Green">E62</font></i></a>, </i>
;<br/><a NAME="E17:66_1_2"/>
 <a href="polynom8.html#K5" target="_self">aConv</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="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p2">(<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="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_9"><b>proof </b></a><div class="add">

<i><font color="Green">E73</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K5" target="_self">aConv</a> <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="matrix_3.html#K6">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span>,<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>)</span> = 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</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><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_9_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:66_1_2_9_2"/>
<font color="Maroon">u'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K5" target="_self">aConv</a> <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="matrix_3.html#K6">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span>,<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>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_9_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_2_9_2_1"/>
( <font color="Maroon">u</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> or 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_9_2_1_1"><b>suppose </b></a><a NAME="E1:66_1_2_9_2_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">u</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><i><font color="Green">E75</font></i>:  <a href="finseq_1.html#K3">len</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="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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 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><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>
<br/>.= 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="polynom8.html#E9:66_1_2"><i><font color="Green">E66</font></i></a>, </i>
;<br/><a NAME="E3:66_1_2_9_2_1_1"/><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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 class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2_9_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:66_1_2_9_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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 class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E9:66_1_2"><i><font color="Green">E66</font></i></a>, </i>;<br/>
<a NAME="E2:66_1_2_9_2_1_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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 class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:66_1_2_9_2_1_1_2"/><i><font color="Green">E79</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:66_1_2_9_2_1_1_2"/>
<font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E3:66_1_2_9_2_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:66_1_2_9_2_1_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:66_1_2_9_2_1_1_2"/><i><font color="Green">E91</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="txt" href="polynom8.html#E9:66_1_2"><i><font color="Green">E66</font></i></a>, </i>;<br/>
<a NAME="E7:66_1_2_9_2_1_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
;<br/>
<b>hence </b><a NAME="E8:66_1_2_9_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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 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>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><a href="polynom8.html#K5" target="_self">aConv</a> <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="matrix_3.html#K6">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span>,<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u'</font> = 
<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="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1
<b>by </b><i>, , <a class="ref" href="polynom8.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p2">(<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="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="polynom8.html#T5" target="_self">Th5</a>, <a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p2">(<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="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i>, </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="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> <a href="funct_1.html#K1">.</a> <font color="Maroon">u'</font>
<b>by </b><i><a class="ref" href="polynom5.html#D3">POLYNOM5:def 3</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_9_2_1_2"><b>suppose </b></a><a NAME="E1:66_1_2_9_2_1_2"/><i><font color="Green">E92</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u</font>
 ;<br/><div class="add"><i><font color="Green">E93</font></i>:  <a href="finseq_1.html#K3">len</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="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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 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><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>
<br/>.= 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="polynom8.html#E9:66_1_2"><i><font color="Green">E66</font></i></a>, </i>
;<br/><a NAME="E3:66_1_2_9_2_1_2"/><i><font color="Green">E94</font></i>: 
 <a href="algseq_1.html#K3">len</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> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="polynom8.html#K5" target="_self">aConv</a> <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="matrix_3.html#K6">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span>,<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>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u'</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i>, , <a class="ref" href="polynom8.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p2">(<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="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p2">(<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="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom8.html#T5" target="_self">Th5</a>, <a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></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="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> <a href="funct_1.html#K1">.</a> <font color="Maroon">u'</font>
<b>by </b><i><a class="ref" href="polynom5.html#D3">POLYNOM5:def 3</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:66_1_2_9"/>
 <a href="polynom8.html#K5" target="_self">aConv</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="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</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>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p2">(<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="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="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E18:66_1_2"/>
 <a href="polynom8.html#K6" target="_self">DFT</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</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>,<font color="Maroon">x</font>,<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"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <span class="p2">(<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="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="txt" href="polynom8.html#E14:66_1_2"><i><font color="Green">E70</font></i></a>, , <a class="txt" href="polynom8.html#E4:66_1_2"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
