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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:24"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="int_2.html#V1">prime</a> )
 ;<br/>

<b>assume </b><a NAME="E2:24"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>assume </b><a NAME="E3:24"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E4:24"/><i><font color="Green">E28</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 ;<br/>

<a NAME="E5:24"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nat_4.html#E3:24"><i><font color="Green">E27</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:24"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/>
<a NAME="E7:24"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:24"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E8:24"/><b>then </b><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:24"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:24"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E10:24"/>
<font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:24"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E11:24"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:24"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:24"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:24"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E15:24"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E14:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nat_4.html#E1:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nat_4.html#E7:24"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E16:24"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E5:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nat_4.html#E14:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nat_4.html#E2:24"><i><font color="Green">E26</font></i></a>, <a class="txt" href="nat_4.html#E15:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="nat_4.html#E3:24"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pepin.html#T9">PEPIN:9</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/>
<a NAME="E17:24"/>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E8:24"><i><font color="Green">E32</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E18:24"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E19:24"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:24"><i><font color="Green">E27</font></i></a>, <a class="txt" href="nat_4.html#E16:24"><i><font color="Green">E37</font></i></a>, <a class="ref" href="nat_4.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E20:24"/><i><font color="Green">E39</font></i>: 
 <a href="pepin.html#K4">order</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E12:24"><i><font color="Green">E34</font></i></a>, <a class="txt" href="nat_4.html#E5:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nat_4.html#E1:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nat_4.html#E7:24"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nat_4.html#E9:24"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nat_4.html#E16:24"><i><font color="Green">E37</font></i></a>, <a class="txt" href="nat_4.html#E19:24"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pepin.html#T52">PEPIN:52</a></i>;<br/>
<a NAME="E21:24"/>
<font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&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">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/>

<a NAME="E2:24_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E3:24_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T12">PREPOWER:12</a></i>;<br/>
<a NAME="E4:24_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E8:24"><i><font color="Green">E32</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E5:24_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default">0 <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="nat_4.html#E3:24_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="nat_4.html#E5:24_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="nat_1.html#T63">NAT_1:63</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">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:24"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E23:24"/><b>then </b>
not <font color="Maroon">c<sub>6</sub></font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="scpinvar.html#T2">SCPINVAR:2</a>, <a class="txt" href="nat_4.html#E4:24"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E24:24"/><b>then </b>
not <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E5:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pepin.html#T8">PEPIN:8</a></i>;<br/>
<a NAME="E25:24"/><b>then </b><i><font color="Green">E40</font></i>: 
not  <a href="pepin.html#K4">order</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E5:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nat_4.html#E19:24"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pepin.html#T53">PEPIN:53</a></i>;<br/>
<a NAME="E26:24"/>
<font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:24"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E27:24"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E28:24"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="nat_4.html#E1:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nat_4.html#E8:24"><i><font color="Green">E32</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a>, <a class="txt" href="nat_4.html#E28:24"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>3</sub></font>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E5:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>
;<br/>
<a NAME="E30:24"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#R1">divides</a>  <a href="pepin.html#K4">order</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E25:24"><i><font color="Green">E40</font></i></a>, <a class="txt" href="nat_4.html#E28:24"><i><font color="Green">E41</font></i></a>, <a class="txt" href="nat_4.html#E20:24"><i><font color="Green">E39</font></i></a>, <a class="txt" href="nat_4.html#E1:24"><i><font color="Green">E25</font></i></a>, <a class="ref" href="nat_4.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E31:24"/>
<font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E5:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E32:24"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<a NAME="E33:24"/><i><font color="Green">E44</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nat_4.html#E5:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pepin.html#T26">PEPIN:26</a></i>;<br/>
<a NAME="E34:24"/>
 <a href="pepin.html#K4">order</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:24"><i><font color="Green">E27</font></i></a>, <a class="txt" href="nat_4.html#E19:24"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pepin.html#T54">PEPIN:54</a></i>;<br/>
<a NAME="E35:24"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T51">NAT_1:51</a>, <a class="txt" href="nat_4.html#E30:24"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E36:24"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K4">mod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E6:24"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pepin.html#T6">PEPIN:6</a></i>;<br/>
<b>hence </b><a NAME="E37:24"/>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K4">mod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E32:24"><i><font color="Green">E43</font></i></a>, <a class="txt" href="nat_4.html#E33:24"><i><font color="Green">E44</font></i></a>, <a class="ref" href="nat_1.html#T66">NAT_1:66</a></i>;<br/>


</div>
