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<b>let </b><font color="Maroon" title="c1">q1</font>, <font color="Maroon" title="c2">q</font>, <font color="Maroon" title="c3">n1</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">q1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c3">n1</font> &amp; <font color="Maroon" title="c2">q</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c1">q1</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c3">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q1</font> )</span><br/>

<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c1">q1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">q</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c3">n1</font> &amp; <font color="Maroon" title="c2">q</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">q1</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> or  not <font color="Maroon" title="c3">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q1</font> )</span><br/>

<b>assume </b><a NAME="E2:19"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c1">q1</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c3">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q1</font></span><br/>

<a NAME="E3:19"/><b>then </b><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c1">q1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<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>then consider </b><font color="Maroon" title="c4">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:19"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c1">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:19"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="pepin.html#T33" title="PEPIN:th.33">PEPIN:33</a></i>;<br/>
<a NAME="E6:19"/>
<font color="Maroon" title="c4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">q1</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E5:19"><i><font color="Green" title="E23">A4</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="E7:19"/><b>then </b><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q1</font> )
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E8:19"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">q1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:19_1"/><b>then </b>
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="nat_4.html#E5:19"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="xcmplx_1.html#T7" title="XCMPLX_1:th.7">XCMPLX_1:7</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E1:19"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:19"/>
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q1</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E5:19"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="nat_4.html#E7:19"><i><font color="Green" title="E24">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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