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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="vectsp_2.html#NM1" title="VECTSP_2:NM.1">comRing</a>;<br/><b>let </b><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</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>let </b><font color="Maroon" title="c4">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<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><span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> $1</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> $1</span>)</span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10" title="POLYNOM3:func.10">1_.</a> <font color="Maroon" title="c1">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T16" title="POLYNOM5:th.16">POLYNOM5:16</a></i>
<br/>.= 
<font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T36" title="POLYNOM3:th.36">POLYNOM3:36</a></i>
;<br/>
<a NAME="E2:45"/><b>then </b><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:45"/><i><font color="Green" title="E36">A2</font></i>: 
for <font color="Olive" title="b1">j</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">j</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">j</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="E1:45_2"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font>]
 ;<br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T20" title="POLYNOM5:th.20">POLYNOM5:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c4">p</font>
<b>by </b><i><a class="txt" href="uproots.html#E1:45_2"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="polynom3.html#T34" title="POLYNOM3:th.34">POLYNOM3:34</a></i>
<br/>.= 
<font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T20" title="POLYNOM5:th.20">POLYNOM5:20</a></i>
<br/>.= 
<font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E3:45_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:45"/>
for <font color="Olive" title="b1">j</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">j</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#E2:45"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="uproots.html#E3:45"><i><font color="Green" title="E36">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:45"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c3">j</font></span>)</span>
 ;<br/>


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