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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means  <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">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></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="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>;<br/>
<a NAME="E1:54"/>
<span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> 0 <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K13">1_.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="polynom5.html#T16">POLYNOM5:16</a></i>;<br/>
<a NAME="E2:54"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="polynom4.html#T7">POLYNOM4:7</a></i>;<br/>
<a NAME="E3:54"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:54_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <font color="Maroon">c<sub>4</sub></font> as  <a href="uproots.html#V1" target="_self">non-zero</a> <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="uproots.html#E1:54_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="uproots.html#T19" target="_self">Th19</a></i>;<br/>
<b>thus </b> <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">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom5.html#K2">`^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></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"><a href="polynom5.html#K4">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></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> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#T32" target="_self">Th32</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">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T17">POLYNOM5:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="txt" href="uproots.html#E1:54_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="uproots.html#T40" target="_self">Th40</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:54"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="uproots.html#E2:54"><i><font color="Green">E42</font></i></a>, <a class="txt" href="uproots.html#E3:54"><i><font color="Green">E43</font></i></a>);</i><br/>


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