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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E34">A1</font></i>: 
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">p</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 ;<br/>

<a NAME="E2:47"/><b>then </b><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E3:47"/><i><font color="Green" title="E36">A3</font></i>: 
( <font color="Maroon" title="c3">n</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#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <font color="Maroon" title="c3">n</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#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">k</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1"/><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c3">n</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#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_1_1"/>
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  ex <font color="Olive" title="b1">i</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="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Olive" title="b1">i</font> )
 </span><b>by </b><i><a class="txt" href="pepin.html#E1:47"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="pepin.html#E1:47_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="pepin.html#T33" target="_self" title="PEPIN:th.33">Th33</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1_1"><b>suppose </b></a><a NAME="E1:47_1_1_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:47_1_1_1_1"/>
<font color="Maroon" title="c3">n</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#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1_2"><b>suppose </b></a><a NAME="E1:47_1_1_1_2"/>
 ex <font color="Olive" title="b1">i</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="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Olive" title="b1">i</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">i</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="E3:47_1_1_1_2"/><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">i</font>
 ;<br/><b>consider </b><font color="Maroon" title="c5">u</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:47_1_1_1_2"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">u</font>
 <b>by </b><i><a class="txt" href="pepin.html#E1:47_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="pepin.html#E3:47_1_1_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E6:47_1_1_1_2"/>
<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="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">u</font></span>)</span>
 <b>by </b><i><a class="txt" href="pepin.html#E5:47_1_1_1_2"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/><a NAME="E7:47_1_1_1_2"/><b>then </b><i><font color="Green" title="E40">A7</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">k</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="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">u</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#E2:47"><i><font color="Green" title="E35">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="E8:47_1_1_1_2"/><b>then </b><i><font color="Green" title="E41">A8</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">u</font>
 <b>by </b><i><a class="txt" href="pepin.html#E2:47"><i><font color="Green" title="E35">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="E9:47_1_1_1_2"/><b>then </b><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c5">u</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#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">k</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="E10:47_1_1_1_2"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="pepin.html#E1:47"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="pepin.html#E3:47_1_1_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="pepin.html#E8:47_1_1_1_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">j</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="E12:47_1_1_1_2"/><i><font color="Green" title="E43">A10</font></i>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c6">j</font>
 <b>by </b><i><a class="txt" href="pepin.html#E1:47"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="pepin.html#E9:47_1_1_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="pepin.html#T33" target="_self" title="PEPIN:th.33">Th33</a></i>;<br/><a NAME="E13:47_1_1_1_2"/>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c6">j</font>
 <b>by </b><i><a class="txt" href="pepin.html#E2:47"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="pepin.html#E3:47_1_1_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="pepin.html#E7:47_1_1_1_2"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="pepin.html#E12:47_1_1_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="nat_d.html#T18" title="NAT_D:th.18">NAT_D:18</a></i>;<br/><b>hence </b><a NAME="E14:47_1_1_1_2"/>
<font color="Maroon" title="c3">n</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#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">k</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><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:47_1"/>
<font color="Maroon" title="c3">n</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#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">k</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:47"/>
( <font color="Maroon" title="c3">n</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#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">k</font> implies <font color="Maroon" title="c3">n</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#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_2"/>
<font color="Maroon" title="c3">n</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#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">k</font>
 ;<br/>

<a NAME="E2:47_2"/><b>then </b>
<font color="Maroon" title="c3">n</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">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/>
<b>hence </b><a NAME="E3:47_2"/>
<font color="Maroon" title="c3">n</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#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:47"/>
( <font color="Maroon" title="c3">n</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#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> iff <font color="Maroon" title="c3">n</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#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">k</font> )
 <b>by </b><i><a class="txt" href="pepin.html#E3:47"><i><font color="Green" title="E36">A3</font></i></a></i>;<br/>


</div>
