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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="vectsp_1.html#V4">right-distributive</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="binom.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>;<br/>
<a NAME="E1:49"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>by </b><i><a class="ref" href="binom.html#T8">BINOM:8</a></i>;<br/>
<a NAME="E2:49"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:49_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/>

<b>assume </b><a NAME="E2:49_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font>]
 ;<br/>

<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="binom.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="binom.html#K2">|^</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="binom.html#T9">BINOM:9</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/>
<b>hence </b><a NAME="E4:49_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:49"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(, <a class="ref" href="polynom8.html#T4" target="_self">Th4</a>);<br/></i>
<b>assume </b><a NAME="E4:49"/><i><font color="Green">E42</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> <a href="polynom8.html#R1" target="_self">is_primitive_root_of_degree</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E5:49"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="polynom8.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:49"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
;<br/>
<a NAME="E7:49"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E8:49"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E9:49"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="binom.html#K2">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:49"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="binom.html#K2">|^</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</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>hence </b><a NAME="E11:49"/>
contradiction
 <b>by </b><i>, <a class="ref" href="polynom8.html#T3" target="_self">Th3</a></i>;<br/>


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