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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="polynom5.html#V2" title="POLYNOM5:attr.2">algebraic-closed</a>  <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="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>  st  <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#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 holds <br/> <a href="polyvie1.html#K2" title="POLYVIE1:func.2">SumRoots</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a>  <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"> (  <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#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 implies  <a href="polyvie1.html#K2" title="POLYVIE1:func.2">SumRoots</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><span class="p4">(<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_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><span class="p4">(<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_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:41"/>
 <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#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <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="polyvie1.html#K2" title="POLYVIE1:func.2">SumRoots</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><span class="p4">(<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_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><span class="p4">(<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_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E2:41"/><span class="kw">then </span>
(  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 
;<br/>
<a NAME="E3:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 not  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T28" onmouseover="rs('nat_2/T28')" onmouseout="rh()">NAT_2:28</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">p</font> being   <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>  st $1 <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="Olive" title="b1">p</font> holds <br/> <a href="polyvie1.html#K2" title="POLYVIE1:func.2">SumRoots</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default">$1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default">$1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E4:41"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[2]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a>  <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"> ( 2 <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="c3">p</font> implies  <a href="polyvie1.html#K2" title="POLYVIE1:func.2">SumRoots</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default">2 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default">2 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_1"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <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="polyvie1.html#K2" title="POLYVIE1:func.2">SumRoots</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default">2 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default">2 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c5">b</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:41_1"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyvie1.html#T6" target="_self" onmouseover="rs('polyvie1/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E4:41_1"/>
( <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="polynom5.html#T38" onmouseover="rs('polynom5/T38')" onmouseout="rh()">POLYNOM5:38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:41_1"/>
 <a href="polyvie1.html#K2" title="POLYVIE1:func.2">SumRoots</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default">2 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default">2 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="polyvie1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="txt" href="polyvie1.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a>, <a class="ref" href="polyvie1.html#T27" target="_self" onmouseover="rs('polyvie1/T27')" onmouseout="rh()">Th27</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="E5:41"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
 for <font color="Olive" title="b1">k</font> being   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">k</font> be   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <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">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_2"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a>  <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"> ( <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c4">p</font> implies  <a href="polyvie1.html#K2" title="POLYVIE1:func.2">SumRoots</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:41_2"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c4">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">  <a href="polyvie1.html#K2" title="POLYVIE1:func.2">SumRoots</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E3:41_2"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" onmouseover="rs('nat_d/T34')" onmouseout="rh()">NAT_D:34</a></span>;<br/>
<a NAME="E4:41_2"/>
<font color="Maroon" title="c3">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> 2 <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>, <a class="ref" href="nat_2.html#T29" onmouseover="rs('nat_2/T29')" onmouseout="rh()">NAT_2:29</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E30">A8</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">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> 2
<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/>.= 
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">k1</font> = <font color="Maroon" title="c3">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> ;<br/>
<a NAME="E7:41_2"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T29" onmouseover="rs('nat_2/T29')" onmouseout="rh()">NAT_2:29</a></span>;<br/>
<a NAME="E8:41_2"/>
<font color="Maroon" title="c3">k</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> <font color="Maroon" title="c3">k</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></span>;<br/>
<a NAME="E9:41_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">k</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> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:41_2"><span class="lab"><font color="Green" title="E31">A9</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="E10:41_2"/><span class="kw">then </span>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:41_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</font> is  <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D9" onmouseover="rs('polynom5/D9')" onmouseout="rh()">POLYNOM5:def 9</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E13:41_2"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<font color="Maroon" title="c6">r</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c4">p</font>
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">P</font> =  <a href="uproots.html#K5" title="UPROOTS:func.5">poly_quotient</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font>);<br/>
<a NAME="E14:41_2"/><span class="lab"><font color="Green" title="E33">A11</font></span>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K5" title="UPROOTS:func.5">poly_quotient</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font>)</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c4">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E13:41_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="uproots.html#D7" onmouseover="rs('uproots/D7')" onmouseout="rh()">UPROOTS:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">P</font> =  <a href="uproots.html#K5" title="UPROOTS:func.5">poly_quotient</a> (<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">r</font>) as   <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="uproots.html#T17" onmouseover="rs('uproots/T17')" onmouseout="rh()">UPROOTS:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">k2</font> = <font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 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="nat_2.html#T29" onmouseover="rs('nat_2/T29')" onmouseout="rh()">NAT_2:29</a>, <a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/>
<a NAME="E17:41_2"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k2</font>
 
<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="E18:41_2"/><span class="lab"><font color="Green" title="E35">A13</font></span>: 
<font color="Maroon" title="c3">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="c5">k1</font>
 
<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="E19:41_2"/><span class="lab"><font color="Green" title="E36">A14</font></span>: 
<font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c3">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="#E2:41_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E14:41_2"><span class="lab"><font color="Green" title="E33">A11</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 NAME="E20:41_2"/><span class="lab"><font color="Green" title="E37">A15</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c8">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:41_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="uproots.html#T50" onmouseover="rs('uproots/T50')" onmouseout="rh()">UPROOTS:50</a></span>;<br/>
<a NAME="E21:41_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A16</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k1</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"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T37" onmouseover="rs('uproots/T37')" onmouseout="rh()">UPROOTS:37</a></span>;<br/>
<a NAME="E22:41_2"/><span class="lab"><font color="Green" title="E39">A17</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k2</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"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">k2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_2"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="uproots.html#T37" onmouseover="rs('uproots/T37')" onmouseout="rh()">UPROOTS:37</a></span>;<br/>
<a NAME="E23:41_2"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">r</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T9" onmouseover="rs('vectsp_1/T9')" onmouseout="rh()">VECTSP_1:9</a></span>;<br/>
<a NAME="E24:41_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A18</font></span>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">k2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T30" onmouseover="rs('rlvect_1/T30')" onmouseout="rh()">RLVECT_1:30</a></span>;<br/>
<a NAME="E25:41_2"/><span class="lab"><font color="Green" title="E41">A19</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E14:41_2"><span class="lab"><font color="Green" title="E33">A11</font></span></a></span>;<br/>
<a NAME="E26:41_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A20</font></span>: 
<font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algseq_1.html#T10" onmouseover="rs('algseq_1/T10')" onmouseout="rh()">ALGSEQ_1:10</a></span>;<br/>
<a NAME="E27:41_2"/><span class="lab"><font color="Green" title="E43">A21</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K9" title="ALGSTR_0:func.9">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K9" title="ALGSTR_0:func.9">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span>
 
;<br/>
<a NAME="E28:41_2"/>
 not <font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:41_2"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="ref" href="algseq_1.html#T10" onmouseover="rs('algseq_1/T10')" onmouseout="rh()">ALGSEQ_1:10</a></span>;<br/>
<a NAME="E29:41_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A22</font></span>: 
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:41_2"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="ref" href="polyvie1.html#T4" target="_self" onmouseover="rs('polyvie1/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">thus </span> <a href="polyvie1.html#K2" title="POLYVIE1:func.2">SumRoots</a> <font color="Maroon" title="c4">p</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polyvie1.html#K2" title="POLYVIE1:func.2">SumRoots</a> <font color="Maroon" title="c8">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_2"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="ref" href="polyvie1.html#T28" target="_self" onmouseover="rs('polyvie1/T28')" onmouseout="rh()">Th28</a></span>
<br/>.= 
<font color="Maroon" title="c6">r</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">k2</font></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_2"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E14:41_2"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="txt" href="#E17:41_2"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E18:41_2"><span class="lab"><font color="Green" title="E35">A13</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c9">k2</font></span>)</span></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c5">k1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:41_2"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E29:41_2"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="ref" href="vectsp_2.html#T20" onmouseover="rs('vectsp_2/T20')" onmouseout="rh()">VECTSP_2:20</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E5:41_2"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="txt" href="#E19:41_2"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E21:41_2"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="txt" href="#E22:41_2"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="txt" href="#E24:41_2"><span class="lab"><font color="Green" title="E40">A18</font></span></a>, <a class="txt" href="#E26:41_2"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="ref" href="vectsp_2.html#T19" onmouseover="rs('vectsp_2/T19')" onmouseout="rh()">VECTSP_2:19</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:41"/>
 for <font color="Olive" title="b1">k</font> being   non  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_2.html#S2" onmouseover="rs('nat_2/S2')" onmouseout="rh()">NAT_2:sch 2</a>(<a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E26">A4</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E7:41"/>
 <a href="polyvie1.html#K2" title="POLYVIE1:func.2">SumRoots</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><span class="p4">(<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_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p3">(<span class="default"><span class="p4">(<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_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E24">A1</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>
