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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b2">m</font> being    <a href="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="b2">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="b2">m</font> holds <br/> for <font color="Olive" title="b3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b4">i</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="b4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">m</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">i</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#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being    <a href="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> 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><br/>  for <font color="Olive" title="b3">i</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">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">m</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">i</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#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="c3">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="c3">m</font> implies  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b2">i</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="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">i</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#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
<font color="Maroon" title="c3">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="E2:64"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b2">i</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="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">i</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#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</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">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">v</font> =  <a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>);<br/>
<a NAME="E3:64"/>
 for <font color="Olive" title="b1">i</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">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:64_1"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</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="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> implies <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_1"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">k</font></span><br/>

<a NAME="E2:64_1_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">k</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> <font color="Maroon" title="c3">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E3:64_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A5</font></span>: 
<font color="Maroon" title="c6">k</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> <font color="Maroon" title="c3">m</font>
 
;<br/>
<a NAME="E4:64_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> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</a></span>;<br/>
<a NAME="E5:64_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">k</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#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> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:64_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</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="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:64_1_1"/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</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>
<span class="lab"><font color="Green" title="E61">A6</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> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> = 
<font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E62">A7</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
;<br/>
 <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="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> = 
 <a href="matrix_0.html#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> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1"><span class="lab"><font color="Green" title="E61">A6</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="#E1:64"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</a></span>
;<br/>
<a NAME="E5:64_1"/><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="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> is   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c3">m</font>,1,<font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E3:64_1"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="ref" href="matrix_0.html#T20" onmouseover="rs('matrix_0/T20')" onmouseout="rh()">MATRIX_0:20</a></span>;<br/>
<a NAME="E6:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A8</font></span>: 
 <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p5">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T25" onmouseover="rs('matrix_0/T25')" onmouseout="rh()">MATRIX_0:25</a></span>;<br/>
<span class="lab"><font color="Green" title="E64">A9</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> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> = 
<font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E58">A1</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="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_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="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> = 
 <a href="matrix_0.html#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> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1"><span class="lab"><font color="Green" title="E61">A6</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="#E1:64"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</a></span>
;<br/>
<a NAME="E9:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A10</font></span>: 
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"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span></span>)</span>
 
;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">i</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="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> implies <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</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#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1) )</span><br/>

<span class="kw">assume </span><a NAME="E10:64_1"/><span class="lab"><font color="Green" title="E66">A11</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</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#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1)</span><br/>

<a NAME="E11:64_1"/><span class="lab"><font color="Green" title="E67">A12</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">k</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="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> implies <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_7"/>
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span><br/>

<a NAME="E2:64_1_7"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c7">k</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> <font color="Maroon" title="c3">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E3:64_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c7">k</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> <font color="Maroon" title="c3">m</font>
 
;<br/>
<a NAME="E4:64_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c7">k</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"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></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">hence </span><a NAME="E5:64_1_7"/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</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="E12:64_1"/><span class="lab"><font color="Green" title="E68">A13</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">k</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="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> implies <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_8"/><span class="lab"><font color="Green" title="E69">A14</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span>)</span><br/>

<a NAME="E2:64_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A15</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c7">k</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> <font color="Maroon" title="c3">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E3:64_1_8"/><span class="lab"><font color="Green" title="E71">A16</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">i</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> <font color="Maroon" title="c3">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:64_1"><span class="lab"><font color="Green" title="E67">A12</font></span></a>, <a class="txt" href="#E1:64_1_8"><span class="lab"><font color="Green" title="E69">A14</font></span></a></span>
<br/>.= 
 <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_8"><span class="lab"><font color="Green" title="E71">A16</font></span></a>, <a class="txt" href="#E2:64_1_8"><span class="lab"><font color="Green" title="E70">A15</font></span></a>, <a class="ref" href="polynom8.html#D8" target="_self" onmouseover="rs('polynom8/D8')" onmouseout="rh()">Def7</a></span>
<br/>.= 
 <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span>)

;<br/>
<span class="kw">hence </span><a NAME="E5:64_1_8"/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:64_1"/><span class="lab"><font color="Green" title="E69">A17</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><font color="Olive" title="b2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b4">c</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">k</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">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b3">c</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c10">c</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="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> implies <font color="Maroon" title="c8">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">c</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64_1_9"/><span class="lab"><font color="Green" title="E70">A18</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:64_1_9"/><span class="lab"><font color="Green" title="E71">A19</font></span>: 
( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c7">k</font> )
 <span class="kw">and </span><br/><a NAME="E3:64_1_9"/><span class="lab"><font color="Green" title="E72">A20</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">c</font></span><br/>

<a NAME="E4:64_1_9"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:64_1"><span class="lab"><font color="Green" title="E68">A13</font></span></a>, <a class="txt" href="#E1:64_1_9"><span class="lab"><font color="Green" title="E70">A18</font></span></a>, <a class="txt" href="#E3:64_1_9"><span class="lab"><font color="Green" title="E72">A20</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:64_1_9"/>
<font color="Maroon" title="c8">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E2:64_1_9"><span class="lab"><font color="Green" title="E71">A19</font></span></a>, <a class="txt" href="#E3:64_1_9"><span class="lab"><font color="Green" title="E72">A20</font></span></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="E14:64_1"/><span class="lab"><font color="Green" title="E70">A21</font></span>: 
 for <font color="Olive" title="b1">k</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_10"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:64_1_10"/><span class="lab"><font color="Green" title="E71">A22</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> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">k</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> implies <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:64_1_10"/><span class="lab"><font color="Green" title="E72">A23</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">and </span><br/><a NAME="E3:64_1_10"/><span class="lab"><font color="Green" title="E73">A24</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><br/>

<a NAME="E4:64_1_10"/>
<font color="Maroon" title="c7">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_10"><span class="lab"><font color="Green" title="E72">A23</font></span></a>, <a class="txt" href="#E3:64_1_10"><span class="lab"><font color="Green" title="E73">A24</font></span></a></span>;<br/>
<a NAME="E5:64_1_10"/><span class="kw">then </span>
<font color="Maroon" title="c7">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_10"><span class="lab"><font color="Green" title="E71">A22</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:64_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A25</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c7">k</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/>
<a NAME="E7:64_1_10"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_10"><span class="lab"><font color="Green" title="E72">A23</font></span></a></span>;<br/>
<a NAME="E8:64_1_10"/><span class="kw">then </span>
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is    <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>, <a class="ref" href="nat_1.html#T20" onmouseover="rs('nat_1/T20')" onmouseout="rh()">NAT_1:20</a></span>;<br/>
<a NAME="E9:64_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A26</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E10:64_1_10"/>
 <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#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">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> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>;<br/>
<a NAME="E11:64_1_10"/><span class="kw">then </span>
<font color="Maroon" title="c7">k</font> <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"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_10"><span class="lab"><font color="Green" title="E72">A23</font></span></a>, <a class="txt" href="#E3:64_1_10"><span class="lab"><font color="Green" title="E73">A24</font></span></a></span>;<br/>
<a NAME="E12:64_1_10"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c7">k</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">a</font> = <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>, <font color="Maroon" title="c9">b</font> = <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>, <font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_1_10"><span class="lab"><font color="Green" title="E74">A25</font></span></a>, <a class="txt" href="#E9:64_1_10"><span class="lab"><font color="Green" title="E75">A26</font></span></a></span>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>
;<br/>
<span class="kw">then </span> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</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="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#T6" onmouseover="rs('matrix_3/T6')" onmouseout="rh()">MATRIX_3:6</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="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
;<br/>
<a NAME="E16:64_1_10"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</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> <font color="Maroon" title="c3">m</font>
 
<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="E17:64_1_10"/><span class="kw">then </span>
<font color="Maroon" title="c7">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_10"><span class="lab"><font color="Green" title="E72">A23</font></span></a>, <a class="txt" href="#E3:64_1_10"><span class="lab"><font color="Green" title="E73">A24</font></span></a></span>;<br/>
<a NAME="E18:64_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A27</font></span>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T60" onmouseover="rs('fvsum_1/T60')" onmouseout="rh()">FVSUM_1:60</a></span>;<br/>
<a NAME="E19:64_1_10"/><span class="lab"><font color="Green" title="E77">A28</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">c</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_10_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:64_1_10_3"/><span class="lab"><font color="Green" title="E78">A29</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_10"><span class="lab"><font color="Green" title="E72">A23</font></span></a></span>;<br/>
<a NAME="E2:64_1_10_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A30</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" onmouseover="rs('nat_1/T20')" onmouseout="rh()">NAT_1:20</a></span>;<br/>
<a NAME="E3:64_1_10_3"/>
( <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E3:64_1_10"><span class="lab"><font color="Green" title="E73">A24</font></span></a>, <a class="ref" href="nat_1.html#T20" onmouseover="rs('nat_1/T20')" onmouseout="rh()">NAT_1:20</a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<a NAME="E4:64_1_10_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A31</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_10_3"><span class="lab"><font color="Green" title="E79">A30</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">l</font> = <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_10_3"><span class="lab"><font color="Green" title="E78">A29</font></span></a>, <a class="ref" href="nat_1.html#T20" onmouseover="rs('nat_1/T20')" onmouseout="rh()">NAT_1:20</a></span>;<br/>
<a NAME="E6:64_1_10_3"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E7:64_1_10_3"/>
<font color="Maroon" title="c8">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:64_1"><span class="lab"><font color="Green" title="E69">A17</font></span></a>, <a class="txt" href="#E4:64_1_10_3"><span class="lab"><font color="Green" title="E80">A31</font></span></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>
<span class="kw">reconsider </span><font color="Maroon" title="c11">d</font> =  <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>) as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E21:64_1_10"/><span class="lab"><font color="Green" title="E78">A32</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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="#E2:64_1_10"><span class="lab"><font color="Green" title="E72">A23</font></span></a>, <a class="ref" href="nat_1.html#T20" onmouseover="rs('nat_1/T20')" onmouseout="rh()">NAT_1:20</a></span>;<br/>
<font color="Maroon" title="c11">d</font> = 
 <a href="polynom8.html#K2" title="POLYNOM8:func.2">pow</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_10"><span class="lab"><font color="Green" title="E72">A23</font></span></a>, <a class="ref" href="polynom8.html#T27" target="_self" onmouseover="rs('polynom8/T27')" onmouseout="rh()">Th27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:64_1_10"><span class="lab"><font color="Green" title="E78">A32</font></span></a>, <a class="ref" href="polynom8.html#D3" target="_self" onmouseover="rs('polynom8/D3')" onmouseout="rh()">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:64_1_10"><span class="lab"><font color="Green" title="E78">A32</font></span></a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E23:64_1_10"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:64_1_10"><span class="lab"><font color="Green" title="E77">A28</font></span></a>, <a class="txt" href="#E18:64_1_10"><span class="lab"><font color="Green" title="E76">A27</font></span></a>, <a class="txt" href="#E21:64_1_10"><span class="lab"><font color="Green" title="E78">A32</font></span></a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:64_1"/><span class="lab"><font color="Green" title="E71">A33</font></span>: 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_11"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:64_1_11"/><span class="lab"><font color="Green" title="E72">A34</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st  <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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</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>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_11_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:64_1_11_1"/><span class="lab"><font color="Green" title="E73">A35</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <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="c2">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> implies <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</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><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:64_1_11_1"/><span class="lab"><font color="Green" title="E74">A36</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">and </span><br/><a NAME="E3:64_1_11_1"/><span class="lab"><font color="Green" title="E75">A37</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</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><br/>

<a NAME="E4:64_1_11_1"/><span class="lab"><font color="Green" title="E76">A38</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#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_11_1"><span class="lab"><font color="Green" title="E74">A36</font></span></a></span>;<br/>
<a NAME="E5:64_1_11_1"/>
<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> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_11_1"><span class="lab"><font color="Green" title="E74">A36</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E6:64_1_11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A39</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_11_1"><span class="lab"><font color="Green" title="E73">A35</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="E7:64_1_11_1"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<a NAME="E8:64_1_11_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<a NAME="E9:64_1_11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A40</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_11_1"><span class="lab"><font color="Green" title="E76">A38</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E10:64_1_11_1"/>
<font color="Maroon" title="c7">k</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">then </span><span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:64_1"><span class="lab"><font color="Green" title="E70">A21</font></span></a>, <a class="txt" href="#E3:64_1_11_1"><span class="lab"><font color="Green" title="E75">A37</font></span></a>, <a class="txt" href="#E6:64_1_11_1"><span class="lab"><font color="Green" title="E77">A39</font></span></a></span>
<br/>.= 
<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> <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="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:64_1_11_1"><span class="lab"><font color="Green" title="E78">A40</font></span></a>, <a class="ref" href="algseq_1.html#T8" onmouseover="rs('algseq_1/T8')" onmouseout="rh()">ALGSEQ_1:8</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>

;<br/>
<span class="kw">hence </span><a NAME="E12:64_1_11_1"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</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>
 ; <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 href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="polynom8.html#T28" target="_self" onmouseover="rs('polynom8/T28')" onmouseout="rh()">Th28</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E73">A41</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</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="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#T6" onmouseover="rs('matrix_3/T6')" onmouseout="rh()">MATRIX_3:6</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="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
;<br/>
<a NAME="E4:64_1_11"/>
<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">lengthG</font> = <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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> 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="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">F</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E7:64_1_11"/><span class="lab"><font color="Green" title="E74">A42</font></span>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">and </span><br/><a NAME="E8:64_1_11"/><span class="lab"><font color="Green" title="E75">A43</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">and </span><br/><a NAME="E9:64_1_11"/><span class="lab"><font color="Green" title="E76">A44</font></span>: 
 for <font color="Olive" title="b1">k</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">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">F</font> holds <br/><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#D2" onmouseover="rs('polynom4/D2')" onmouseout="rh()">POLYNOM4:def 2</a></span>;<br/>
<a NAME="E10:64_1_11"/>
 ex <font color="Olive" title="b1">G</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font> st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">lengthG</font> &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">lengthG</font> holds <br/><font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</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> ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_11_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>$2 <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>;<br/>
<a NAME="E1:64_1_11_4"/><span class="lab"><font color="Green" title="E77">A45</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">lengthG</font> holds <br/> ex <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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>]
 
;<br/>
<a NAME="E2:64_1_11_4"/>
 ex <font color="Olive" title="b1">G</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font> st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">lengthG</font> &amp; (  for <font color="Olive" title="b2">nn</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">nn</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">lengthG</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">nn</font>,<font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">nn</font>] ) )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" onmouseover="rs('finseq_1/S5')" onmouseout="rh()">FINSEQ_1:sch 5</a>(<a class="txt" href="#E1:64_1_11_4"><span class="lab"><font color="Green" title="E77">A45</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E3:64_1_11_4"/>
 ex <font color="Olive" title="b1">G</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font> st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">lengthG</font> &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">lengthG</font> holds <br/><font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</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> ) )
 ; <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>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">G</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E12:64_1_11"/><span class="lab"><font color="Green" title="E77">A46</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">lengthG</font>
 <span class="kw">and </span><br/><a NAME="E13:64_1_11"/><span class="lab"><font color="Green" title="E78">A47</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">lengthG</font> holds <br/><font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</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>
 ;<br/>
<a NAME="E14:64_1_11"/><span class="lab"><font color="Green" title="E79">A48</font></span>: 
 for <font color="Olive" title="b1">k</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">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">lengthG</font> holds <br/><font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</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="txt" href="#E13:64_1_11"><span class="lab"><font color="Green" title="E78">A47</font></span></a></span>;<br/>
<a NAME="E15:64_1_11"/><span class="lab"><font color="Green" title="E80">A49</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:64_1_11"><span class="lab"><font color="Green" title="E77">A46</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E16:64_1_11"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:64_1_11"><span class="lab"><font color="Green" title="E75">A43</font></span></a></span>;<br/>
<a NAME="E17:64_1_11"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></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> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/>
<a NAME="E18:64_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A50</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_11"><span class="lab"><font color="Green" title="E73">A41</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E19:64_1_11"/><span class="lab"><font color="Green" title="E82">A51</font></span>: 
 for <font color="Olive" title="b1">k</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> holds <br/><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_11_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</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"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> implies <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64_1_11_5"/><span class="lab"><font color="Green" title="E83">A52</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">and </span><br/><a NAME="E2:64_1_11_5"/><span class="lab"><font color="Green" title="E84">A53</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</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="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<a NAME="E3:64_1_11_5"/><span class="lab"><font color="Green" title="E85">A54</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E2:64_1_11_5"><span class="lab"><font color="Green" title="E84">A53</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="E4:64_1_11_5"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">F</font> <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"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:64_1_11"><span class="lab"><font color="Green" title="E75">A43</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:64_1_11_5"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_11_5"><span class="lab"><font color="Green" title="E83">A52</font></span></a>, <a class="txt" href="#E2:64_1_11_5"><span class="lab"><font color="Green" title="E84">A53</font></span></a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></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="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="nat_1.html#K9" title="NAT_1:func.9">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:64_1_11"><span class="lab"><font color="Green" title="E76">A44</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:64_1"><span class="lab"><font color="Green" title="E70">A21</font></span></a>, <a class="txt" href="#E1:64_1_11_5"><span class="lab"><font color="Green" title="E83">A52</font></span></a>, <a class="txt" href="#E3:64_1_11_5"><span class="lab"><font color="Green" title="E85">A54</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:64_1_11_5"/>
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:64_1_11"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_11_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</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="c10">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> implies <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/>

<a NAME="E1:64_1_11_6"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:64_1_11"><span class="lab"><font color="Green" title="E75">A43</font></span></a>, <a class="txt" href="#E15:64_1_11"><span class="lab"><font color="Green" title="E80">A49</font></span></a></span>;<br/>
<a NAME="E2:64_1_11_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A55</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></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> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:64_1_11_6"/><span class="lab"><font color="Green" title="E84">A56</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</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="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_11_6_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="E1:64_1_11_6_1"/>
( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:64_1_11"><span class="lab"><font color="Green" title="E81">A50</font></span></a>, <a class="txt" href="#E3:64_1_11_6"><span class="lab"><font color="Green" title="E84">A56</font></span></a>, <a class="txt" href="#E2:64_1_11_6"><span class="lab"><font color="Green" title="E83">A55</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_11_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_11_6_1_1"/><span class="lab"><font color="Green" title="E85">A57</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</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="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:64_1_11_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:64_1_11"><span class="lab"><font color="Green" title="E75">A43</font></span></a>, <a class="txt" href="#E19:64_1_11"><span class="lab"><font color="Green" title="E82">A51</font></span></a>, <a class="txt" href="#E3:64_1_11_6"><span class="lab"><font color="Green" title="E84">A56</font></span></a>, <a class="txt" href="#E1:64_1_11_6_1_1"><span class="lab"><font color="Green" title="E85">A57</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E4:64_1_11_6_1_1"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_11_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_11_6_1_2"/><span class="lab"><font color="Green" title="E85">A58</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:64_1_11_6_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> <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="c10">k</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="E3:64_1_11_6_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">l</font> = <font color="Maroon" title="c10">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> 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="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><a NAME="E5:64_1_11_6_1_2"/>
<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E6:64_1_11_6_1_2"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">lengthG</font> = <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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> 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="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><a NAME="E8:64_1_11_6_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> <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="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_11_6_1_2"><span class="lab"><font color="Green" title="E85">A58</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E9:64_1_11_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A59</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E10:64_1_11_6_1_2"/>
<font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">lengthG</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:64_1_11"><span class="lab"><font color="Green" title="E75">A43</font></span></a>, <a class="txt" href="#E1:64_1_11_6_1_2"><span class="lab"><font color="Green" title="E85">A58</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E11:64_1_11_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A60</font></span>: 
<font color="Maroon" title="c11">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:64_1_11"><span class="lab"><font color="Green" title="E77">A46</font></span></a>, <a class="txt" href="#E9:64_1_11_6_1_2"><span class="lab"><font color="Green" title="E86">A59</font></span></a></span>;<br/><a NAME="E12:64_1_11_6_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:64_1_11"><span class="lab"><font color="Green" title="E75">A43</font></span></a>, <a class="txt" href="#E15:64_1_11"><span class="lab"><font color="Green" title="E80">A49</font></span></a></span>;<br/><a NAME="E13:64_1_11_6_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></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> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/><a NAME="E14:64_1_11_6_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 <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/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_11_6_1_2"><span class="lab"><font color="Green" title="E85">A58</font></span></a>, <a class="ref" href="finseq_1.html#T24" onmouseover="rs('finseq_1/T24')" onmouseout="rh()">FINSEQ_1:24</a></span>
<br/>.= 
 <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="#E12:64_1_11"><span class="lab"><font color="Green" title="E77">A46</font></span></a>, <a class="txt" href="#E13:64_1_11"><span class="lab"><font color="Green" title="E78">A47</font></span></a>, <a class="txt" href="#E11:64_1_11_6_1_2"><span class="lab"><font color="Green" title="E87">A60</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:64_1_11"><span class="lab"><font color="Green" title="E75">A43</font></span></a>, <a class="txt" href="#E1:64_1_11"><span class="lab"><font color="Green" title="E72">A34</font></span></a>, <a class="txt" href="#E1:64_1_11_6_1_2"><span class="lab"><font color="Green" title="E85">A58</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E16:64_1_11_6_1_2"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</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"> 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 NAME="E21:64_1_11"/><span class="kw">then </span>
 <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:64_1_11"><span class="lab"><font color="Green" title="E81">A50</font></span></a>, <a class="ref" href="finseq_1.html#T13" onmouseover="rs('finseq_1/T13')" onmouseout="rh()">FINSEQ_1:13</a></span>;<br/>
<span class="kw">then </span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T41" onmouseover="rs('rlvect_1/T41')" onmouseout="rh()">RLVECT_1:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:64_1_11"><span class="lab"><font color="Green" title="E77">A46</font></span></a>, <a class="txt" href="#E14:64_1_11"><span class="lab"><font color="Green" title="E79">A48</font></span></a>, <a class="ref" href="polynom3.html#T1" onmouseover="rs('polynom3/T1')" onmouseout="rh()">POLYNOM3:1</a></span>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64_1_11"><span class="lab"><font color="Green" title="E74">A42</font></span></a>, <a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E23:64_1_11"/>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:64_1"/><span class="lab"><font color="Green" title="E72">A61</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> (<span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<span class="p3">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span>,1)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#D9" onmouseover="rs('fvsum_1/D9')" onmouseout="rh()">FVSUM_1:def 9</a></span>;<br/>
<a NAME="E17:64_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">i</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> <font color="Maroon" title="c3">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E18:64_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">i</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> <font color="Maroon" title="c3">m</font>
 
;<br/>
<a NAME="E19:64_1"/><span class="kw">then </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="c6">i</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"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_1"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="txt" href="#E9:64_1"><span class="lab"><font color="Green" title="E65">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E20:64_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">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>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> (<font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64_1"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="txt" href="#E16:64_1"><span class="lab"><font color="Green" title="E72">A61</font></span></a>, <a class="txt" href="#E15:64_1"><span class="lab"><font color="Green" title="E71">A33</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/>
<span class="kw">hence </span><a NAME="E21:64_1"/>
<span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c6">i</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#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</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="#E10:64_1"><span class="lab"><font color="Green" title="E66">A11</font></span></a>, <a class="ref" href="polynom8.html#D7" target="_self" onmouseover="rs('polynom8/D7')" onmouseout="rh()">Def6</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>
<span class="kw">hence </span><a NAME="E4:64"/>
 for <font color="Olive" title="b1">i</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">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K5" title="POLYNOM8:func.5">DFT</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK7" title="POLYNOM8:NK.7">VM</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">mConv</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font>)</span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1)
 ; <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>
