<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L</font> be  non <a href="vectsp_1.html#V10">degenerated</a> <a href="vectsp_2.html#NM1">comRing</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">q</font> be  <a href="uproots.html#V1" target="_self">non-zero</a> <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/>





<b>set </b><font color="Maroon">r</font> = <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span>;<br/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> 0</span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span> = 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T16">POLYNOM5:16</a></i>
<br/>.= 
0 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T36">POLYNOM3:36</a></i>
;<br/>
<a NAME="E2:56"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:56"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:56_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>]
 ;<br/>

<a NAME="E2:56_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E3:56_2"/><b>then </b>
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:56_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <font color="Maroon">i</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font> is <a href="uproots.html#V1" target="_self">non-zero</a>
 
<b>by </b><i><a class="txt" href="uproots.html#E4:56_2"><i><font color="Green">E45</font></i></a>, </i>;<br/>
<b>thus </b> <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span> = 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">x</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> 1</span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">x</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">x</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T17">POLYNOM5:17</a></i>
<br/>.= 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">x</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <font color="Maroon">i</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T34">POLYNOM3:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="txt" href="uproots.html#E4:56_2"><i><font color="Green">E45</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span>

;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:56"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="uproots.html#E3:56"><i><font color="Green">E43</font></i></a>, );</i><br/>


</div>
