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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Olive" title="b2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> <font color="Olive" title="b1">m</font></span><br/><b>let </b><font color="Maroon" title="c2">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> <font color="Maroon" title="c2">m</font> holds <br/><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> <font color="Maroon" title="c2">m</font></span><br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> <font color="Maroon" title="c2">m</font> implies <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> <font color="Maroon" title="c2">m</font> )</span><br/>





<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> <font color="Maroon" title="c2">m</font></span><br/>

<a NAME="E2:50"/><b>then </b><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="polynom8.html#D6" target="_self" title="POLYNOM8:def.6">Def6</a></i>;<br/>
<a NAME="E3:50"/><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="polynom8.html#T30" target="_self" title="POLYNOM8:th.30">Th30</a></i>;<br/>
<a NAME="E4:50"/>
<font color="Maroon" title="c3">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="polynom8.html#D6" target="_self" title="POLYNOM8:def.6">Def6</a></i>;<br/>
<a NAME="E5:50"/><b>then </b><i><font color="Green" title="E50">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="#E3:50"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="polynom8.html#E40"><i><font color="Green" title="E38">Lm10</font></i></a></i>;<br/>
<a NAME="E6:50"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font> )</span><br/>

<b>assume </b><a NAME="E1:50_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E2:50_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="polynom8.html#D6" target="_self" title="POLYNOM8:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E3:50_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="#E3:50"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="polynom8.html#E41"><i><font color="Green" title="E39">Lm11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:50"/>
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  <a href="polynom8.html#R1" title="POLYNOM8:pred.1">is_primitive_root_of_degree</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="#E5:50"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="polynom8.html#D6" target="_self" title="POLYNOM8:def.6">Def6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
