<?xml version="1.0"?>
<div class="add">

<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#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</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">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</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>assume </b><a NAME="E1:51"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="polynom8.html#R1" target="_self">is_primitive_root_of_degree</a> <font color="Maroon">m</font>
 ;<br/>

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:51"/><i><font color="Green">E38</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 ;<br/>

<a NAME="E3:51"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1"><b>suppose </b></a><a NAME="E1:51_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E3:51_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E3:51_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xreal_1.html#T16">XREAL_1:16</a></i>;<br/><a NAME="E4:51_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T15">XREAL_1:15</a></i>;<br/><a NAME="E5:51_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:51_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:51_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E8:51_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">k</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="polynom8.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E9:51_1_1"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <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/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_2"><b>suppose </b></a><a NAME="E1:51_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:51_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:51_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T16">XREAL_1:16</a></i>;<br/><a NAME="E4:51_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="prepower.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> =  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:51_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T16">XREAL_1:16</a></i>;<br/><a NAME="E7:51_1_2"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T15">XREAL_1:15</a></i>;<br/><a NAME="E8:51_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:51_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:51_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E11:51_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> )
 <b>by </b><i>, <a class="ref" href="polynom8.html#T3" target="_self">Th3</a>, , , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1_2_1"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="vectsp_1.html#K4">"</a>  <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
 ;<br/> <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i>, <a class="txt" href="polynom8.html#E4:51_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="vectsp_1.html#D22">VECTSP_1:def 22</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>
;<br/><b>hence </b><a NAME="E3:51_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="polynom8.html#T3" target="_self">Th3</a>, <a class="txt" href="polynom8.html#E3:51_1_2"><i><font color="Green">E55</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:51_1_2"/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <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="txt" href="polynom8.html#E2:51_1_2"><i><font color="Green">E53</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
