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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="vectsp_2.html#NM1" title="VECTSP_2:NM.1">comRing</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">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> <br/> for <font color="Olive" title="b3">p</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span></span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">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#K13" title="POLYNOM3:func.13">*'</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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="Olive" title="b1">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#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">j</font></span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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#K13" title="POLYNOM3:func.13">*'</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#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">j</font></span>)</span></span><br/>





<span class="kw">defpred </span><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> ] <span class="kw">means </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#K13" title="POLYNOM3:func.13">*'</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#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/>
<a NAME="E1:48"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
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#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:48_1"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</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#K13" title="POLYNOM3:func.13">*'</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#K2" title="NAT_1:func.2">+</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#K13" title="POLYNOM3:func.13">*'</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#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T20" title="POLYNOM5:th.20">POLYNOM5:20</a></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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c4">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="polynom3.html#T34" title="POLYNOM3:th.34">POLYNOM3:34</a></span>
<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#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T20" title="POLYNOM5:th.20">POLYNOM5:20</a></span>
<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#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E3:48_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<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#K13" title="POLYNOM3:func.13">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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#K13" title="POLYNOM3:func.13">*'</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T16" title="POLYNOM5:th.16">POLYNOM5:16</a></span>
<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#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T36" title="POLYNOM3:th.36">POLYNOM3:36</a></span>
;<br/>
<a NAME="E3:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E4:48"/>
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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E32">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:48"/>
<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#K13" title="POLYNOM3:func.13">*'</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#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">j</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
