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

<b>let </b><font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <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="E1:50"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 );<br/>
<a NAME="E2:50"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:50"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">m</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>]
 ;<br/>

<a NAME="E2:50_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font>, <font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">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/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="euler_2.html#T10">EULER_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="nat_1.html#T77">NAT_1:77</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="nat_1.html#T78">NAT_1:78</a></i>
;<br/>
<b>hence </b><a NAME="E3:50_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:50_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:50"/>
for <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E5:50"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="newton.html#K2">|^</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/>


</div>
