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<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>







<b>set </b><font color="Maroon">ep</font> =  <a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">eq</font> =  <a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">epq</font> =  <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>,<font color="Maroon">m</font>;<br/>
<i><font color="Green">E31</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="polynom8.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></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"><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>,<font color="Maroon">m</font></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="54_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="E1:54_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"><span class="p2">(<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="polynom8.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></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="54_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_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="54_2_1_1"><b>suppose </b></a><a NAME="E1:54_2_1_1"/><i><font color="Green">E38</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"><b>hence </b><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>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u'</font> = 
 <a href="polynom4.html#K2">eval</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>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom4.html#T27">POLYNOM4:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="normsp_1.html#K2">.</a> <font color="Maroon">u</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">u</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="normsp_1.html#K2">.</a> <font color="Maroon">u</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#K6" target="_self">DFT</a> <font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">m</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#K6" target="_self">DFT</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="polynom8.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u'</font>
<b>by </b><i/>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_2"><b>suppose </b></a><a NAME="E1:54_2_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">u</font>
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<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="polynom8.html#K2" target="_self">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></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#K6" target="_self">DFT</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">u</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#K6" target="_self">DFT</a> <font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <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#K6" target="_self">DFT</a> <font color="Maroon">q</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<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#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>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u'</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:54"/>
<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="polynom8.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">q</font>,<font color="Maroon">x</font>,<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"><font color="Maroon">p</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">x</font>,<font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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