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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> for <font color="Olive" title="b3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds   <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">m</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom8.html#K4" title="POLYNOM8:func.4">aConv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">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#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">m</font>)</span>)</span></span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">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="Olive" title="b1">m</font> holds <br/> for <font color="Olive" title="b2">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds   <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">m</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom8.html#K4" title="POLYNOM8:func.4">aConv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">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#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">m</font>)</span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">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> implies  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds   <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">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#K4" title="POLYNOM8:func.4">aConv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Olive" title="b1">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#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:65"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">and </span><br/><a NAME="E2:65"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
 <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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds   <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">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#K4" title="POLYNOM8:func.4">aConv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Olive" title="b1">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#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span></span><br/>

<span class="kw">let </span><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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="polynom8.html#K5" title="POLYNOM8:func.5">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#K4" title="POLYNOM8:func.4">aConv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span></span><br/>

<a NAME="E3:65"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E62">A4</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">u9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">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> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">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="Olive" title="b1">u9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K3" title="POLYNOM8:func.3">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="Olive" title="b1">u9</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">u9</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">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> implies <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K3" title="POLYNOM8:func.3">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="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:65_1"/>
<font color="Maroon" title="c5">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">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 class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K3" title="POLYNOM8:func.3">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="Olive">b<sub>1</sub></font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">u</font> = <font color="Maroon" title="c5">u9</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:65_1_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_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:65_1_1_1"/><span class="lab"><font color="Green" title="E63">A5</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K3" title="POLYNOM8:func.3">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="Olive">b<sub>1</sub></font></span><br/><div class="add"> <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A6</font></span>:  <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#NK7" title="POLYNOM8:NK.7">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#K3" title="POLYNOM8:func.3">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#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">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">u9</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E1:65_1_1_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="polynom8.html#T41" target="_self" onmouseover="rs('polynom8/T41')" onmouseout="rh()">Th41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K3" title="POLYNOM8:func.3">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">u9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_1"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E3:65_1_1_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="ref" href="polynom8.html#D5" target="_self" onmouseover="rs('polynom8/D5')" onmouseout="rh()">Def4</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:65_1_1_2"/><span class="lab"><font color="Green" title="E63">A7</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K3" title="POLYNOM8:func.3">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="Olive">b<sub>1</sub></font></span><br/><div class="add"> <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A8</font></span>:  <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#NK7" title="POLYNOM8:NK.7">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#K3" title="POLYNOM8:func.3">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#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">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">u9</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_2"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="ref" href="polynom8.html#D7" target="_self" onmouseover="rs('polynom8/D7')" onmouseout="rh()">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K3" title="POLYNOM8:func.3">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">u9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1_2"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="txt" href="#E3:65_1_1_2"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="ref" href="polynom8.html#D5" target="_self" onmouseover="rs('polynom8/D5')" onmouseout="rh()">Def4</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">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#NK1" title="NUMBERS:NK.1">NAT</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:65"/>
 <a href="polynom8.html#K5" title="POLYNOM8:func.5">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#K4" title="POLYNOM8:func.4">aConv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">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#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
