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<div class="add">

<b>let </b><font color="Maroon">L</font> be   <a href="vectsp_2.html#NM2">domRing</a>;<br/><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>let </b><font color="Maroon">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E2:44"/>
( <span class="p1">(<span class="default"><a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font> &amp; <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="polynom3.html#T28">POLYNOM3:28</a>, <a class="ref" href="polynom3.html#T31">POLYNOM3:31</a></i>;<br/>
<a NAME="E3:44"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<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><font color="Maroon">p</font> <a href="polynom5.html#K2">`^</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>;<br/>
<a NAME="E4:44"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, <a class="ref" href="polynom5.html#T16">POLYNOM5:16</a></i>;<br/>
<a NAME="E5:44"/><i><font color="Green">E45</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="44_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="E1:44_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<a NAME="E2:44_1"/>
<font color="Maroon">p</font> <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"><font color="Maroon">p</font> <a href="polynom5.html#K2">`^</a> <font color="Maroon">n</font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="polynom5.html#T20">POLYNOM5:20</a></i>;<br/>
<b>hence </b><a NAME="E3:44_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>, <a class="ref" href="uproots.html#T2" target="_self">Th2</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:44"/>
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>(<a class="ref" href="uproots.html#T1" target="_self">Th1</a>, );<br/></i>
<b>hence </b><a NAME="E7:44"/>
<font color="Maroon">p</font> <a href="polynom5.html#K2">`^</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
 ;<br/>


</div>
