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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">degenerated</a> <a href="vectsp_2.html#NM1" title="VECTSP_2:NM.1">comRing</a>;<br/><b>let </b><font color="Maroon" title="c2">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/><b>let </b><font color="Maroon" title="c3">q</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>;<br/>





<b>set </b><font color="Maroon" title="c4">r</font> = <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span>;<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> <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"><span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> $1</span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span>;<br/>
 <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"><span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> 0</span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> = 
 <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"><a href="polynom3.html#K13" title="POLYNOM3:func.13">1_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T16" title="POLYNOM5:th.16">POLYNOM5:16</a></i>
<br/>.= 
0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T36" title="POLYNOM3:th.36">POLYNOM3:36</a></i>
;<br/>
<a NAME="E2:56"/><b>then </b><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:56"/><i><font color="Green" title="E44">A2</font></i>: 
for <font color="Olive" title="b1">i</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">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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" title="c5">i</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="E1:56_2"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font>]
 ;<br/>

<a NAME="E2:56_2"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp;  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="uproots.html#T19" target="_self" title="UPROOTS:th.19">Th19</a></i>;<br/>
<a NAME="E3:56_2"/><b>then </b>
<font color="Maroon" title="c5">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:56_2"/><b>then </b><i><font color="Green" title="E46">A4</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font> is <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:56_2"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="uproots.html#T19" target="_self" title="UPROOTS:th.19">Th19</a></i>;<br/>
<b>thus </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"><span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> = 
 <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"><span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> 1</span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#T32" target="_self" title="UPROOTS:th.32">Th32</a></i>
<br/>.= 
 <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"><span class="p3"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T17" title="POLYNOM5:th.17">POLYNOM5:17</a></i>
<br/>.= 
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p5">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T34" title="POLYNOM3:th.34">POLYNOM3:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
<b>by </b><i><a class="txt" href="uproots.html#E1:56_2"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="uproots.html#E4:56_2"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="uproots.html#T40" target="_self" title="UPROOTS:th.40">Th40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font></span>)</span>

;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:56"/>
for <font color="Olive" title="b1">i</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">i</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="uproots.html#E2:56"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="uproots.html#E3:56"><i><font color="Green" title="E44">A2</font></i></a>);</i><br/>


</div>
