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

<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">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="E1:45"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c1">p</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>
 ;<br/>

<a NAME="E2:45"/><b>then </b><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<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>(  not <font color="Maroon" title="c2">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> $1 or <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  ex <font color="Olive" title="b1">k</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" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b1">k</font> );<br/>
<a NAME="E3:45"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 0
 ;<br/><a NAME="E2:45_1_1"/><b>then </b>
<font color="Maroon" title="c2">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c4">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:45_1_1"/><i><font color="Green" title="E36">A4</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">t</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>thus </b><a NAME="E5:45_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="txt" href="pepin.html#E4:45_1_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="nat_1.html#T15" title="NAT_1:th.15">NAT_1:15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:45_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:45"/><i><font color="Green" title="E36">A5</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="45_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="E1:45_2"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_2_1"/>
<font color="Maroon" title="c2">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<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>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">u1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:45_2_1"/><i><font color="Green" title="E38">A7</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">u1</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E4:45_2_1"/>
<font color="Maroon" title="c1">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <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="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">u1</font>
 <b>by </b><i><a class="txt" href="pepin.html#E3:45_2_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/><a NAME="E5:45_2_1"/><b>then </b><i><font color="Green" title="E39">A8</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">u1</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_2_1_1_1"/>
( <font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">i</font> or <font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">u1</font> )
 </span><b>by </b><i><a class="txt" href="pepin.html#E1:45"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="pepin.html#E5:45_2_1"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:45_2_1_1_1_1"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">i</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">w1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:45_2_1_1_1_1"/><i><font color="Green" title="E40">A9</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">w1</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E4:45_2_1_1_1_1"/>
<font color="Maroon" title="c6">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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/><b>hence </b><a NAME="E5:45_2_1_1_1_1"/>
<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]
 <b>by </b><i><a class="txt" href="pepin.html#E3:45_2_1_1_1_1"><i><font color="Green" title="E40">A9</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:45_2_1_1_1_2"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">u1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">w2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:45_2_1_1_1_2"/><i><font color="Green" title="E40">A10</font></i>: 
<font color="Maroon" title="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">w2</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E4:45_2_1_1_1_2"/>
<font color="Maroon" title="c1">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <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="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">w2</font></span>)</span>
 <b>by </b><i><a class="txt" href="pepin.html#E3:45_2_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="pepin.html#E3:45_2_1_1_1_2"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/><a NAME="E5:45_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">w2</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="pepin.html#E2:45"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="nat_d.html#T18" title="NAT_D:th.18">NAT_D:18</a></i>;<br/><a NAME="E6:45_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">w2</font>
 <b>by </b><i><a class="txt" href="pepin.html#E2:45"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="nat_d.html#T18" title="NAT_D:th.18">NAT_D:18</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">w3</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> <b> such that </b><br/><a NAME="E8:45_2_1_1_1_2"/><i><font color="Green" title="E41">A11</font></i>: 
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">w3</font> or <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="pepin.html#E1:45_2"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>thus </b><a NAME="E9:45_2_1_1_1_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]
 <b>by </b><i><a class="txt" href="pepin.html#E8:45_2_1_1_1_2"><i><font color="Green" title="E41">A11</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:45_2_1"/>
<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>
<b>hence </b><a NAME="E3:45_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="E5:45"/>
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#E3:45"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="pepin.html#E4:45"><i><font color="Green" title="E36">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:45"/>
(  not <font color="Maroon" title="c2">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c3">n</font> or <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  ex <font color="Olive" title="b1">k</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" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b1">k</font> )
 ;<br/>


</div>
