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<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">right_unital</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a> <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">Field-like</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon" title="c4">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16" title="POLYNOM3:func.16">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span><b> such that </b><br/><a NAME="E2:61_1_1"/><i><font color="Green" title="E43">A1</font></i>: 
 <a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c2">p</font>,<span class="p1"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c4">F</font>
 <b>and </b><br/><a NAME="E3:61_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font>
 <b>and </b><br/><a NAME="E4:61_1_1"/><i><font color="Green" title="E44">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c4">F</font> holds <br/><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</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="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom5.html#D5" target="_self" title="POLYNOM5:def.5">Def5</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>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>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span> holds <br/> <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1;<br/><a NAME="E5:61_1_1"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E1:61_1_1_1"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 0</span>)</span>
 ;<br/>

<a NAME="E2:61_1_1_1"/>
<font color="Maroon" title="c4">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16" title="POLYNOM3:func.16">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
;<br/>
<b>then </b><font color="Maroon" title="c5">p</font> = 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p1">(<span class="default"><a href="polynom3.html#K16" title="POLYNOM3:func.16">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E1:61_1_1_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="rlvect_1.html#T60" title="RLVECT_1:th.60">RLVECT_1:60</a></i>
<br/>.= 
 <a href="polynom3.html#K12" title="POLYNOM3:func.12">0_.</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>
;<br/>
<a NAME="E4:61_1_1_1"/><b>then </b>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="polynom4.html#T6" title="POLYNOM4:th.6">POLYNOM4:6</a></i>;<br/>
<b>hence </b><a NAME="E5:61_1_1_1"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/>


</div><b>end;</b></div><a NAME="E6:61_1_1"/><i><font color="Green" title="E46">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:61_1_1_2"/><i><font color="Green" title="E47">A6</font></i>: 
for <font color="Olive" title="b1">q</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">n</font></span>)</span> holds <br/> <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/>

<b>let </b><font color="Maroon" title="c6">q</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E2:61_1_1_2"/><i><font color="Green" title="E48">A7</font></i>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">F1</font> =  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">n</font></span>)</span> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">maxFq</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F1</font></span>)</span>,<span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T1" title="FINSEQ_2:th.1">FINSEQ_2:1</a></i>;<br/>
<a NAME="E5:61_1_1_2"/><i><font color="Green" title="E49">A8</font></i>: 
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="polynom5.html#E1:61_1_1_2"><i><font color="Green" title="E47">A6</font></i></a></i>;<br/>
<a NAME="E6:61_1_1_2"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_1_2_1_1"/>
( <font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font> or <font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_2_1_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>
 ;<br/><div class="add"><b>then </b> <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> = 
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T26" target="_self" title="POLYNOM5:th.26">Th26</a></i>
<br/>.= 
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p1"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">n</font></span>)</span></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T37" target="_self" title="POLYNOM5:th.37">Th37</a></i>
;<br/><b>hence </b><a NAME="E3:61_1_1_2_1_1_1"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="algseq_1.html#D6" title="ALGSEQ_1:def.6">ALGSEQ_1:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_2_1_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_2_1_1_2"/><b>then </b>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="polynom5.html#T25" target="_self" title="POLYNOM5:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E3:61_1_1_2_1_1_2"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:61_1_1_2"/><b>then </b><i><font color="Green" title="E50">A9</font></i>: 
<font color="Maroon" title="c8">maxFq</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="polynom5.html#E5:61_1_1_2"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="square_1.html#T50" title="SQUARE_1:th.50">SQUARE_1:50</a></i>;<br/>
<a NAME="E8:61_1_1_2"/><i><font color="Green" title="E51">A10</font></i>: 
( <font color="Maroon" title="c8">maxFq</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F1</font> &amp; <font color="Maroon" title="c8">maxFq</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_1_2_2_1"/>
( <font color="Maroon" title="c5">n</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> or <font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_2_2_1_1"/><i><font color="Green" title="E52">A11</font></i>: 
<font color="Maroon" title="c5">n</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_2_2_1_1"/>
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:61_1_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E53">A12</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E1:61_1_1_2_2_1_1"><i><font color="Green" title="E52">A11</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E54">A13</font></i>: <font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E4:61_1_1"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="polynom5.html#E3:61_1_1_2_2_1_1"><i><font color="Green" title="E53">A12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
;<br/><a NAME="E5:61_1_1_2_2_1_1"/>
<font color="Maroon" title="c4">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</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"><font color="Maroon" title="c4">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom5.html#E1:61_1_1_2_2_1_1"><i><font color="Green" title="E52">A11</font></i></a>, <a class="ref" href="jordan8.html#T2" title="JORDAN8:th.2">JORDAN8:2</a></i>;<br/><b>then </b><font color="Maroon" title="c6">q</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E2:61_1_1_2"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="fvsum_1.html#T87" title="FVSUM_1:th.87">FVSUM_1:87</a></i>
<br/>.= 
<font color="Maroon" title="c7">F1</font> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E4:61_1_1_2_2_1_1"><i><font color="Green" title="E54">A13</font></i></a>, <a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>
;<br/><a NAME="E7:61_1_1_2_2_1_1"/><b>then </b>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c6">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">maxFq</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E8:61_1_1_2"><i><font color="Green" title="E51">A10</font></i></a>, <a class="ref" href="polynom4.html#T9" title="POLYNOM4:th.9">POLYNOM4:9</a></i>;<br/><b>hence </b><a NAME="E8:61_1_1_2_2_1_1"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c6">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="polynom5.html#E7:61_1_1_2"><i><font color="Green" title="E50">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_2_2_1_2"/><i><font color="Green" title="E52">A14</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:61_1_1_2_2_1_2"/><b>then </b>
( <font color="Maroon" title="c4">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c4">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</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="c4">F</font> )
 <b>by </b><i><a class="txt" href="polynom5.html#E1:61_1_1_2_2_1_2"><i><font color="Green" title="E52">A14</font></i></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/><b>hence </b><a NAME="E4:61_1_1_2_2_1_2"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c6">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="polynom5.html#E1:61_1_1_2"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="polynom5.html#E2:61_1_1_2"><i><font color="Green" title="E48">A7</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:61_1_1_2"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c6">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/>


</div><b>end;</b></div><a NAME="E7:61_1_1"/><i><font color="Green" title="E47">A15</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="polynom5.html#E5:61_1_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="polynom5.html#E6:61_1_1"><i><font color="Green" title="E46">A5</font></i></a>);<br/></i><a NAME="E8:61_1_1"/>
<font color="Maroon" title="c4">F</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/><b>hence </b><a NAME="E9:61_1_1"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c2">p</font>,<span class="p2"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="polynom5.html#E2:61_1_1"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="polynom5.html#E7:61_1_1"><i><font color="Green" title="E47">A15</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:61_1"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c2">p</font>,<span class="p2"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:61"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom5.html#K5" title="POLYNOM5:func.5">Subst</a> <font color="Maroon" title="c2">p</font>,<span class="p2"><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="algseq_1.html#K5" title="ALGSEQ_1:func.5">%&gt;</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/>


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