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

<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">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> ;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> $1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> $1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font>;<br/>
<a NAME="E1:17"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<a NAME="E1:17_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 0</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/>
<b>hence </b><a NAME="E2:17_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:17"/><i><font color="Green" title="E14">A2</font></i>: 
for <font color="Olive" title="b1">n</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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">n</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> ;<br/>

<b>assume </b><a NAME="E1:17_2"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ;<br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="euler_2.html#T11" title="EULER_2:th.11">EULER_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="txt" href="pepin.html#E1:17_2"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="euler_2.html#T10" title="EULER_2:th.10">EULER_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/>
<b>hence </b><a NAME="E3:17_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:17"/>
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="pepin.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="pepin.html#E2:17"><i><font color="Green" title="E14">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:17"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">m</font>
 ;<br/>


</div>
