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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">d</font>, <font color="Maroon" title="c3">k</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="E1:51"/><i><font color="Green" title="E39">A1</font></i>: 
( <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="c2">d</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c2">d</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">k</font> &amp; not <font color="Maroon" title="c2">d</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#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">p</font> )
 ;<br/>

<a NAME="E2:51"/><b>then </b><i><font color="Green" title="E40">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/>
<a NAME="E3:51"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E2:51_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<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="E3:51_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pepin.html#E1:51"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:51"/><b>then </b><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E5:51"/><b>then </b><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">t</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="E7:51"/><i><font color="Green" title="E43">A5</font></i>: 
( <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c4">t</font> &amp; <font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="txt" href="pepin.html#E1:51"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="pepin.html#T35" target="_self" title="PEPIN:th.35">Th35</a></i>;<br/>
<a NAME="E8:51"/>
not <font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_2"/>
<font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 ;<br/>

<a NAME="E2:51_2"/><b>then </b>
<font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
;<br/>
<a NAME="E3:51_2"/><b>then </b>
<font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E5:51"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E4:51_2"/><b>then </b>
<font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:51_2"/><b>then </b>
<font color="Maroon" title="c2">d</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="c3">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="pepin.html#E7:51"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="pepin.html#E18"><i><font color="Green" title="E14">Lm6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:51_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pepin.html#E1:51"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="pepin.html#E2:51"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="pepin.html#E4:51"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="pepin.html#T29" target="_self" title="PEPIN:th.29">Th29</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:51"/>
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="pepin.html#E7:51"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div>
