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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>





<a NAME="E1:65"/><i><font color="Green" title="E61">A1</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:65"/><i><font color="Green" title="E62">A2</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<i><font color="Green" title="E63">A3</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" title="POLYNOM8:func.6">DFT</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> = 
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">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" title="c5">u'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:65_2"/>
<font color="Maroon" title="c5">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" title="POLYNOM8:func.6">DFT</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">u</font> = <font color="Maroon" title="c5">u'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">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" title="c6">u</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">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" title="E64">A4</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 ;<br/><div class="add"> <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> = 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:65"><i><font color="Green" title="E62">A2</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>
;<br/><b>then </b><i><font color="Green" title="E65">A5</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="polynom8.html#K6" title="POLYNOM8:func.6">DFT</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1
<b>by </b><i><a class="txt" href="polynom8.html#E1:65"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="polynom8.html#E2:65"><i><font color="Green" title="E62">A2</font></i></a>, <a class="txt" href="polynom8.html#E1:65_2_1_1"><i><font color="Green" title="E64">A4</font></i></a>, <a class="ref" href="polynom8.html#T41" target="_self" title="POLYNOM8:th.41">Th41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u'</font>
<b>by </b><i><a class="txt" href="polynom8.html#E1:65_2_1_1"><i><font color="Green" title="E64">A4</font></i></a>, <a class="txt" href="polynom8.html#E3:65_2_1_1"><i><font color="Green" title="E65">A5</font></i></a>, <a class="ref" href="polynom8.html#D5" target="_self" title="POLYNOM8:def.5">Def5</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" title="E64">A6</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">u</font>
 ;<br/><div class="add"> <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> = 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:65"><i><font color="Green" title="E62">A2</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>
;<br/><b>then </b><i><font color="Green" title="E65">A7</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="polynom8.html#K6" title="POLYNOM8:func.6">DFT</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u'</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="polynom8.html#E1:65_2_1_2"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="polynom8.html#D7" target="_self" title="POLYNOM8:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">u'</font>
<b>by </b><i><a class="txt" href="polynom8.html#E1:65_2_1_2"><i><font color="Green" title="E64">A6</font></i></a>, <a class="txt" href="polynom8.html#E3:65_2_1_2"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="polynom8.html#D5" target="_self" title="POLYNOM8:def.5">Def5</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" title="POLYNOM8:func.6">DFT</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom8.html#K5" title="POLYNOM8:func.5">aConv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E3:65"><i><font color="Green" title="E63">A3</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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