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





<a NAME="E1:65"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">m</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>assume </b><a NAME="E2:65"/><i><font color="Green">E38</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &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> )
 ;<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/>

<i><font color="Green">E39</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></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"><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#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></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="65_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="E1:65_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#K6" target="_self">DFT</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></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="65_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_2_1"/>
( <font color="Maroon">u</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> or <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="65_2_1_1"><b>suppose </b></a><a NAME="E1:65_2_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">u</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"> <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>,<font color="Maroon">m</font></span>)</span> = 
<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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i>, </i>
;<br/><b>then </b><i><font color="Green">E51</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>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></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>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></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#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></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>, , , </i>
<br/>.= 
<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#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u'</font>
<b>by </b><i>, , <a class="ref" href="polynom8.html#T2" target="_self">Th2</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_2_1_2"><b>suppose </b></a><a NAME="E1:65_2_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u</font>
 ;<br/><div class="add"> <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>,<font color="Maroon">m</font></span>)</span> = 
<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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i>, </i>
;<br/><b>then </b><i><font color="Green">E55</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>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></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>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></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>, </i>
<br/>.= 
<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#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u'</font>
<b>by </b><i>, , <a class="ref" href="polynom8.html#T2" target="_self">Th2</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:65"/>
 <a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font> <a href="funct_2.html#R2">=</a>  <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> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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