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<b>let </b><font color="Maroon">L</font> be   <a href="vectsp_2.html#NM2">domRing</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>set </b><font color="Maroon">r</font> = <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<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="uproots.html#K4" target="_self">degree</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<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></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:81"/><i><font color="Green">E42</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="polynom4.html#T7">POLYNOM4:7</a></i>;<br/>
<a NAME="E2:81"/>
<span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<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> <a href="polynom5.html#K2">`^</a> 0 <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="polynom5.html#T16">POLYNOM5:16</a></i>;<br/>
<a NAME="E3:81"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:81"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">n</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">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:81_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<a NAME="E2:81_1"/><i><font color="Green">E46</font></i>: 
(  <a href="card_1.html#K4">card</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>,1 <a href="uproots.html#K2" target="_self">-bag</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/>
<a NAME="E3:81_1"/>
<span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<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> <a href="polynom5.html#K2">`^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<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">n</font></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<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>
 
<b>by </b><i><a class="ref" href="polynom5.html#T20">POLYNOM5:20</a></i>;<br/>
<b>then </b> <a href="uproots.html#K4" target="_self">degree</a> <span class="p1">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<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">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="uproots.html#K4" target="_self">degree</a> <span class="p2">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <span class="p3">(<span class="default"><span class="p4"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<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">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K4" target="_self">degree</a> <span class="p2">(<span class="default"><a href="uproots.html#K8" target="_self">BRoots</a> <span class="p3"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<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></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K4" target="_self">degree</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>,1 <a href="uproots.html#K2" target="_self">-bag</a> </span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E5:81_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:81"/>
for <font color="Olive">n</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">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


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