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

<b>let </b><font color="Maroon">p</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:66"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>

<a NAME="E2:66"/><b>then </b>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 0
 
;<br/>
<a NAME="E3:66"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
;<br/>
<b>reconsider </b><font color="Maroon">P</font> =  <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">p</font> as   <a href="funcsdom.html#NM1">Ring</a> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">p</font> is   <a href="newton.html#NM1">Prime</a>
 ;<br/><a NAME="E2:66_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">P</font> holds <br/> ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font> st <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/>

<b>assume </b><a NAME="E1:66_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">P</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">p</font> ;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E4:66_1_1"/><i><font color="Green">E40</font></i>: 
(  <a href="int_2.html#K1">abs</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="int_2.html#K1">abs</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E5:66_1_1"/>
( 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 
;<br/>
<a NAME="E6:66_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">a'</font> &amp; 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E7:66_1_1"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a'</font> &amp; <font color="Olive">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> holds <br/><font color="Olive">m</font> <a href="nat_1.html#R1">divides</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:66_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:66_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E4:66_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a'</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_1_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:66_1_1_1_1_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="txt" href="int_3.html#E3:66_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:66_1_1_1_1_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:66_1_1_1_1_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E3:66_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E4:66_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">a'</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E5:66_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:66_1_1_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:66_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<b>hence </b><a NAME="E6:66_1_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> 1
 <b>by </b><i>, , <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:66_1_1"/><b>then </b>
<font color="Maroon">a'</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E9:66_1_1"/><b>then </b>
<font color="Maroon">a'</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_3.html#E12:66_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">t</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E11:66_1_1"/><i><font color="Green">E50</font></i>: 
1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="int_3.html#E7"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<a NAME="E12:66_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">s</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">s</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a></i>;<br/>
<a NAME="E13:66_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">s</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">p</font> <b>by </b><i>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font> ;<br/>
<b>reconsider </b><font color="Maroon">b2</font> = <font color="Maroon">b'</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">a2</font> = <font color="Maroon">a'</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">e</font> = 1 as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b2</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i><a class="txt" href="int_3.html#E6"><i><font color="Green">Lemma30</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">e</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">P</font>
<b>by </b><i>, <a class="txt" href="int_3.html#E6:66_1_1"><i><font color="Green">E44</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E20:66_1_1"/>
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font> st <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">P</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:66_1"/>
 <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">p</font> is  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="rlvect_1.html#V3">Abelian</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="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> 
 <b>by </b><i>, , <a class="ref" href="vectsp_1.html#D20">VECTSP_1:def 20</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2"/><i><font color="Green">E54</font></i>: 
 <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">p</font> is  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="rlvect_1.html#V3">Abelian</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="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/><a NAME="E2:66_2"/>
for <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   holds <br/>(  not <font color="Olive">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> or <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:66_2_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:66_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E4:66_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_2_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:66_2_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:66_2_1_1_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:66_2_1_1_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E3:66_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="txt" href="int_3.html#E3:66_2_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E4:66_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E5:66_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="int_3.html#E1:66_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="int_3.html#E3:66_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="int_3.html#E4:66_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:66_2_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:66_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_2_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:66_2_1_2_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:66_2_1_2_1_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:66_2_1_2_1_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E3:66_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="txt" href="int_3.html#E4:66_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/><a NAME="E4:66_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E5:66_2_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="int_3.html#E3:66_2_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="int_3.html#E4:66_2_1"><i><font color="Green">E56</font></i></a>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:66_2_1_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_3_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_3_1_1"><b>case </b></a><a NAME="E1:66_2_1_3_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_3_1_1"/><b>then </b>
1 <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:66_2_1_3_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_3_1_2"><b>case </b></a><a NAME="E1:66_2_1_3_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_3_1_2"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_2_1_3_1_2_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_3_1_2_1_1_1"><b>case </b></a><a NAME="E1:66_2_1_3_1_2_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_3_1_2_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><a NAME="E3:66_2_1_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/><b>hence </b><a NAME="E4:66_2_1_3_1_2_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_2_1_3_1_2_1_1_2"><b>case </b></a><a NAME="E1:66_2_1_3_1_2_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:66_2_1_3_1_2_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:66_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">n2</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">p</font> <b>by </b><i><a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">n'</font> = <font color="Maroon">n2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">p</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">k2</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">p</font> <b>by </b><i><a class="txt" href="int_3.html#E3:66_2_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">p</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font>, <font color="Maroon">k</font> = <font color="Maroon">k</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E8:66_2_1_3_1_2_1_1_2"/>
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, </i>;<br/><a NAME="E9:66_2_1_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">p</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><font color="Maroon">n'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">k'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k2</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
0
<b>by </b><i>, , </i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E11:66_2_1_3_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="int_3.html#E4:66_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="vectsp_1.html#T44">VECTSP_1:44</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:66_2_1_3_1_2"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:66_2_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:66_2"/>
<font color="Maroon">p</font> is   <a href="newton.html#NM1">Prime</a>
 <b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:66"/>
(  <a href="int_3.html#K9" target="_self">INT.Ring</a> <font color="Maroon">p</font> is  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="rlvect_1.html#V3">Abelian</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="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>  iff <font color="Maroon">p</font> is   <a href="newton.html#NM1">Prime</a> )
 <b>by </b><i/>;<br/>


</div>
