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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>, <font color="Olive" title="b2">q</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="b3">m</font> being    <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>   st <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &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="b3">m</font> &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b2">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> for <font color="Olive" title="b4">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b4">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Olive" title="b2">q</font></span>)</span>,<font color="Olive" title="b4">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b3">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b4">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b3">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b3">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Olive" title="b2">q</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &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> &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</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>  st <font color="Olive" title="b2">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Olive" title="b2">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b1">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b1">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b1">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">m</font> be    <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> ; <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="c4">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &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="c4">m</font> &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">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>  st <font color="Olive" title="b1">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font> holds <br/> <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:66"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:66"/><span class="lab"><font color="Green" title="E61">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="c4">m</font> &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">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>  st <font color="Olive" title="b1">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font> holds <br/> <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">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"> ( <font color="Maroon" title="c5">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font> implies  <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:66"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">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">  <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_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="E1:66_1"/>
(  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or (  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1"><span class="kw">suppose </span></a><a NAME="E1:66_1_1"/><span class="lab"><font color="Green" title="E63">A4</font></span>: 
(  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_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="E1:66_1_1_1"/>
(  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:66_1_1_1_1"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span><br/><div class="add"><a NAME="E2:66_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T5" onmouseover="rs('polynom4/T5')" onmouseout="rh()">POLYNOM4:5</a></span>;<br/><a NAME="E3:66_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T34" onmouseover="rs('polynom3/T34')" onmouseout="rh()">POLYNOM3:34</a></span>;<br/><a NAME="E4:66_1_1_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">L</font> &amp;  <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom8.html#T33" target="_self" onmouseover="rs('polynom8/T33')" onmouseout="rh()">Th33</a>, <a class="ref" href="polynom5.html#T28" onmouseover="rs('polynom5/T28')" onmouseout="rh()">POLYNOM5:28</a></span>;<br/><span class="kw">hence </span><a NAME="E5:66_1_1_1_1"/>
 <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom8.html#T33" target="_self" onmouseover="rs('polynom8/T33')" onmouseout="rh()">Th33</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="66_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:66_1_1_1_2"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span><br/><div class="add"><a NAME="E2:66_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T5" onmouseover="rs('polynom4/T5')" onmouseout="rh()">POLYNOM4:5</a></span>;<br/><a NAME="E3:66_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T34" onmouseover="rs('polynom3/T34')" onmouseout="rh()">POLYNOM3:34</a></span>;<br/><a NAME="E4:66_1_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">L</font> &amp;  <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom3.html#K8" title="POLYNOM3:func.8">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom8.html#T33" target="_self" onmouseover="rs('polynom8/T33')" onmouseout="rh()">Th33</a>, <a class="ref" href="polynom5.html#T28" onmouseover="rs('polynom5/T28')" onmouseout="rh()">POLYNOM5:28</a></span>;<br/><span class="kw">hence </span><a NAME="E5:66_1_1_1_2"/>
 <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom8.html#T33" target="_self" onmouseover="rs('polynom8/T33')" onmouseout="rh()">Th33</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2"><span class="kw">suppose </span></a><a NAME="E1:66_1_2"/><span class="lab"><font color="Green" title="E63">A5</font></span>: 
(  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c6">v1</font> =  <a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c5">x</font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>);<br/><span class="kw">set </span><font color="Maroon" title="c7">v2</font> =  <a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>);<br/><a NAME="E2:66_1_2"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E3:66_1_2"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_2"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="polynom8.html#T9" target="_self" onmouseover="rs('polynom8/T9')" onmouseout="rh()">Th9</a></span>;<br/><a NAME="E4:66_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A7</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66_1_2"><span class="lab"><font color="Green" title="E64">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:66_1_2"/><span class="lab"><font color="Green" title="E66">A8</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font> holds <br/><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="c5">x</font>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="66_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">i</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"> ( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font> implies <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="c5">x</font>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">i</font> )</span><br/>

<a NAME="E1:66_1_2_1"/>
<font color="Maroon" title="c8">i</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/>
<span class="kw">hence </span><a NAME="E2:66_1_2_1"/>
( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font> implies <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="c5">x</font>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c8">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:66_1_2"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="ref" href="polynom8.html#T41" target="_self" onmouseover="rs('polynom8/T41')" onmouseout="rh()">Th41</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><a NAME="E6:66_1_2"/>
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="ref" href="polynom8.html#D7" target="_self" onmouseover="rs('polynom8/D7')" onmouseout="rh()">Def6</a></span>;<br/><a NAME="E7:66_1_2"/><span class="kw">then </span>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font> <a href="algseq_1.html#R1" title="ALGSEQ_1:pred.1">is_at_least_length_of</a>  <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algseq_1.html#D2" onmouseover="rs('algseq_1/D2')" onmouseout="rh()">ALGSEQ_1:def 2</a></span>;<br/><a NAME="E8:66_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A9</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algseq_1.html#D3" onmouseover="rs('algseq_1/D3')" onmouseout="rh()">ALGSEQ_1:def 3</a></span>;<br/><span class="lab"><font color="Green" title="E68">A10</font></span>:  <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> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> = 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">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#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>
<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 NAME="E10:66_1_2"/><span class="lab"><font color="Green" title="E69">A11</font></span>: 
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E60">A1</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E70">A12</font></span>: <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="polynom8.html#T40" target="_self" onmouseover="rs('polynom8/T40')" onmouseout="rh()">Th40</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K3" title="MATRIX_1:func.3">1.</a> (<font color="Maroon" title="c1">L</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="polynom8.html#T39" target="_self" onmouseover="rs('polynom8/T39')" onmouseout="rh()">Th39</a></span>
;<br/><div><span class="lab"><font color="Green" title="E71">A13</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="66_1_2_4"><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#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span></span>)</span> holds <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#K1" title="POLYNOM8:func.1">emb</a> (<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></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="c8">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="c8">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#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span></span>)</span> implies <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#K1" title="POLYNOM8:func.1">emb</a> (<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></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="E1:66_1_2_4"/>
<font color="Maroon" title="c8">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#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span></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#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></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="c9">u</font> = <font color="Maroon" title="c8">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="66_1_2_4_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="E1:66_1_2_4_1"/>
( <font color="Maroon" title="c9">u</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font> or 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">u</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:66_1_2_4_1_1"/><span class="lab"><font color="Green" title="E72">A14</font></span>: 
<font color="Maroon" title="c9">u</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">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#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></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 NAME="E2:66_1_2_4_1_1"/><span class="kw">then </span>
( <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c9">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:66_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A15</font></span>: 
<font color="Maroon" title="c9">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>
 ;<br/><a NAME="E4:66_1_2_4_1_1"/>
 <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> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66_1_2"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</a></span>;<br/><a NAME="E5:66_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A16</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:66_1_2_4_1_1"/>
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 &amp; 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66_1_2"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</a></span>;<br/><a NAME="E7:66_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1_2_4_1_1"><span class="lab"><font color="Green" title="E74">A16</font></span></a>, <a class="txt" href="#E3:66_1_2_4_1_1"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/> <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#K1" title="POLYNOM8:func.1">emb</a> (<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>
<br/>.= 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66_1_2"><span class="lab"><font color="Green" title="E69">A11</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">hence </span><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#K1" title="POLYNOM8:func.1">emb</a> (<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">u9</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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="c9">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="#E1:66_1_2_4_1_1"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="ref" href="polynom8.html#D5" target="_self" onmouseover="rs('polynom8/D5')" onmouseout="rh()">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c9">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:66_1_2_4_1_1"><span class="lab"><font color="Green" title="E75">A17</font></span></a>, <a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_1_2_4_1_1"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">u9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D4" onmouseover="rs('polynom5/D4')" onmouseout="rh()">POLYNOM5:def 4</a></span>
;<br/> <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="66_1_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:66_1_2_4_1_2"/><span class="lab"><font color="Green" title="E72">A18</font></span>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">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#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></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="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#K1" title="POLYNOM8:func.1">emb</a> (<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>
<br/>.= 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66_1_2"><span class="lab"><font color="Green" title="E69">A11</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">hence </span><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#K1" title="POLYNOM8:func.1">emb</a> (<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="#E1:66_1_2_4_1_2"><span class="lab"><font color="Green" title="E72">A18</font></span></a>, <a class="ref" href="polynom8.html#D5" target="_self" onmouseover="rs('polynom8/D5')" onmouseout="rh()">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:66_1_2"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="txt" href="#E1:66_1_2_4_1_2"><span class="lab"><font color="Green" title="E72">A18</font></span></a>, <a class="ref" href="algseq_1.html#T8" onmouseover="rs('algseq_1/T8')" onmouseout="rh()">ALGSEQ_1:8</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">u9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D4" onmouseover="rs('polynom5/D4')" onmouseout="rh()">POLYNOM5:def 4</a></span>
;<br/> <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#K4" title="POLYNOM8:func.4">aConv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span></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"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></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/><a NAME="E14:66_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A19</font></span>: 
 <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#K1" title="POLYNOM8:func.1">emb</a> (<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:66_1_2"><span class="lab"><font color="Green" title="E71">A13</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>;<br/><span class="lab"><font color="Green" title="E73">A20</font></span>:  <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> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> = 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66_1_2"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</a></span>
<br/>.= 
 <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="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>
<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">then </span><span class="lab"><font color="Green" title="E74">A21</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="c5">x</font>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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/>.= 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">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="matrix_0.html#K1" title="MATRIX_0:func.1">width</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="c5">x</font>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span> = 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:66_1_2"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66_1_2"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</a></span>
;<br/><a NAME="E18:66_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> is   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font>,1,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66_1_2"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="txt" href="#E16:66_1_2"><span class="lab"><font color="Green" title="E74">A21</font></span></a>, <a class="ref" href="matrix_0.html#T20" onmouseover="rs('matrix_0/T20')" onmouseout="rh()">MATRIX_0:20</a></span>;<br/><span class="kw">then </span> <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> (<span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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> (<span class="p3">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span> = 
 <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> (<span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</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="c5">x</font>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66_1_2"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="txt" href="#E5:66_1_2"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="ref" href="polynom8.html#T29" target="_self" onmouseover="rs('polynom8/T29')" onmouseout="rh()">Th29</a></span>
<br/>.= 
 <a href="polynom8.html#K4" title="POLYNOM8:func.4">aConv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c5">x</font>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></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> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:66_1_2"><span class="lab"><font color="Green" title="E68">A10</font></span></a>, <a class="txt" href="#E15:66_1_2"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="ref" href="matrix_3.html#T33" onmouseover="rs('matrix_3/T33')" onmouseout="rh()">MATRIX_3:33</a></span>
<br/>.= 
 <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#K1" title="POLYNOM8:func.1">emb</a> (<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K3" title="MATRIX_1:func.3">1.</a> (<font color="Maroon" title="c1">L</font>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</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> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66_1_2"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="txt" href="#E11:66_1_2"><span class="lab"><font color="Green" title="E70">A12</font></span></a>, <a class="ref" href="polynom8.html#T6" target="_self" onmouseover="rs('polynom8/T6')" onmouseout="rh()">Th6</a></span>
<br/>.= 
 <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#K1" title="POLYNOM8:func.1">emb</a> (<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="matrix_3.html#K15" title="MATRIX_3:func.15">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="polynom8.html#E56"><span class="lab"><font color="Green" title="E52">Lm12</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E20:66_1_2"/>
 <a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">x</font>,<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> (<span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">m</font></span>)</span>,<font color="Maroon" title="c1">L</font>)</span>)</span> <a href="polynom5.html#K4" title="POLYNOM5:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66_1_2"><span class="lab"><font color="Green" title="E69">A11</font></span></a>, <a class="txt" href="#E8:66_1_2"><span class="lab"><font color="Green" title="E67">A9</font></span></a>, <a class="txt" href="#E14:66_1_2"><span class="lab"><font color="Green" title="E72">A19</font></span></a>, <a class="ref" href="polynom8.html#T42" target="_self" onmouseover="rs('polynom8/T42')" onmouseout="rh()">Th42</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>

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