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

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">n1</font>, <font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">a</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> ;<br/>

<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c5">n1</font></span>)</span> &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">q</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> )
 ;<br/>

<b>assume </b><a NAME="E2:24"/><i><font color="Green" title="E26">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>assume </b><a NAME="E3:24"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c6">p</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c6">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> )
 ;<br/>

<b>assume </b><a NAME="E4:24"/><i><font color="Green" title="E28">A4</font></i>: 
not <font color="Maroon" title="c6">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 ;<br/>

<a NAME="E5:24"/><i><font color="Green" title="E29">A5</font></i>: 
( <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c3">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:24"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="nat_4.html#E3:24"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:24"/><b>then </b><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/>
<a NAME="E7:24"/><b>then </b><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c5">n1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:24"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E8:24"/><b>then </b><i><font color="Green" title="E32">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:24"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:24"/><b>then </b><i><font color="Green" title="E33">A9</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E10:24"/>
<font color="Maroon" title="c5">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:24"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E11:24"/><b>then </b>
<font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E12:24"/><b>then </b><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c5">n1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E14:24"/><i><font color="Green" title="E35">A11</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:24"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c8">i</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E16:24"/><i><font color="Green" title="E36">A12</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E14:24"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="nat_4.html#E1:24"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="nat_4.html#E7:24"><i><font color="Green" title="E31">A7</font></i></a></i>;<br/>
<a NAME="E17:24"/><i><font color="Green" title="E37">A13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E5:24"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="nat_4.html#E14:24"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="nat_4.html#E2:24"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="nat_4.html#E16:24"><i><font color="Green" title="E36">A12</font></i></a>, <a class="txt" href="nat_4.html#E3:24"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="pepin.html#T9" title="PEPIN:th.9">PEPIN:9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">nn</font> = <font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/>
<a NAME="E18:24"/>
0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E8:24"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E19:24"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E20:24"/><b>then </b><i><font color="Green" title="E38">A14</font></i>: 
<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">p</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:24"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="nat_4.html#E17:24"><i><font color="Green" title="E37">A13</font></i></a>, <a class="ref" href="nat_4.html#T15" target="_self" title="NAT_4:th.15">Th15</a></i>;<br/>
<a NAME="E21:24"/><i><font color="Green" title="E39">A15</font></i>: 
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E12:24"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="nat_4.html#E5:24"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="nat_4.html#E1:24"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="nat_4.html#E7:24"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="nat_4.html#E9:24"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="nat_4.html#E17:24"><i><font color="Green" title="E37">A13</font></i></a>, <a class="txt" href="nat_4.html#E20:24"><i><font color="Green" title="E38">A14</font></i></a>, <a class="ref" href="pepin.html#T52" title="PEPIN:th.52">PEPIN:52</a></i>;<br/>
<a NAME="E22:24"/>
<font color="Maroon" title="c7">a</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon" title="c7">a</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/>

<a NAME="E2:24_1"/><b>then </b>
<font color="Maroon" title="c7">a</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E3:24_1"/><b>then </b><i><font color="Green" title="E40">A16</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/>
<a NAME="E4:24_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E8:24"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E5:24_1"/><b>then </b><i><font color="Green" title="E41">A17</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> = 
<span class="p1">(<span class="default">0 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="nat_4.html#E3:24_1"><i><font color="Green" title="E40">A16</font></i></a>, <a class="txt" href="nat_4.html#E5:24_1"><i><font color="Green" title="E41">A17</font></i></a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="nat_d.html#T13" title="NAT_D:th.13">NAT_D:13</a></i>
;<br/>
<b>hence </b><a NAME="E7:24_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E2:24"><i><font color="Green" title="E26">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:24"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E24:24"/><b>then </b>
not <font color="Maroon" title="c6">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 
<b>by </b><i><a class="ref" href="prepower.html#T109" title="PREPOWER:th.109">PREPOWER:109</a>, <a class="txt" href="nat_4.html#E4:24"><i><font color="Green" title="E28">A4</font></i></a></i>;<br/>
<a NAME="E25:24"/><b>then </b>
not <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E5:24"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="pepin.html#T8" title="PEPIN:th.8">PEPIN:8</a></i>;<br/>
<a NAME="E26:24"/><b>then </b><i><font color="Green" title="E40">A18</font></i>: 
not  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">q</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E5:24"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="nat_4.html#E20:24"><i><font color="Green" title="E38">A14</font></i></a>, <a class="ref" href="pepin.html#T53" title="PEPIN:th.53">PEPIN:53</a></i>;<br/>
<a NAME="E27:24"/>
<font color="Maroon" title="c5">n1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:24"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E28:24"/><b>then </b>
<font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E29:24"/><b>then </b><i><font color="Green" title="E41">A19</font></i>: 
<font color="Maroon" title="c5">n1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">q</font>
<b>by </b><i><a class="txt" href="nat_4.html#E1:24"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="nat_4.html#E8:24"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="txt" href="nat_4.html#E29:24"><i><font color="Green" title="E41">A19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">q</font>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">q</font>

<br/>.= 
<font color="Maroon" title="c2">k</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E5:24"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="nat_d.html#T18" title="NAT_D:th.18">NAT_D:18</a></i>
;<br/>
<a NAME="E31:24"/><b>then </b><i><font color="Green" title="E42">A20</font></i>: 
<font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c5">n1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">p</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E26:24"><i><font color="Green" title="E40">A18</font></i></a>, <a class="txt" href="nat_4.html#E29:24"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="nat_4.html#E21:24"><i><font color="Green" title="E39">A15</font></i></a>, <a class="txt" href="nat_4.html#E1:24"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="nat_4.html#T17" target="_self" title="NAT_4:th.17">Th17</a></i>;<br/>
<a NAME="E32:24"/>
<font color="Maroon" title="c6">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E5:24"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E33:24"/><b>then </b><i><font color="Green" title="E43">A21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font>
 
;<br/>
<a NAME="E34:24"/><i><font color="Green" title="E44">A22</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c5">n1</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:24"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="nat_4.html#E5:24"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="pepin.html#T26" title="PEPIN:th.26">PEPIN:26</a></i>;<br/>
<a NAME="E35:24"/>
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:24"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="nat_4.html#E20:24"><i><font color="Green" title="E38">A14</font></i></a>, <a class="ref" href="pepin.html#T54" title="PEPIN:th.54">PEPIN:54</a></i>;<br/>
<a NAME="E36:24"/><b>then </b>
<font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c5">n1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a>, <a class="txt" href="nat_4.html#E31:24"><i><font color="Green" title="E42">A20</font></i></a></i>;<br/>
<a NAME="E37:24"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c5">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E6:24"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="pepin.html#T6" title="PEPIN:th.6">PEPIN:6</a></i>;<br/>
<b>hence </b><a NAME="E38:24"/>
<font color="Maroon" title="c6">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c5">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E33:24"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="nat_4.html#E34:24"><i><font color="Green" title="E44">A22</font></i></a>, <a class="ref" href="nat_d.html#T16" title="NAT_D:th.16">NAT_D:16</a></i>;<br/>


</div>
