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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="gr_cy_3.html#V1" title="GR_CY_3:attr.1">Safe</a>  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 7 implies <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">k</font> = <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">q</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E2:6"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_3.html#D1" target="_self" title="GR_CY_3:def.1">Def1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:6"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 7
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2</span><br/>

<div><span class="lab"><font color="Green" title="E8">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:6_1"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
( <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:6_1_1_1"/>
( <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:6_1_1_1_1"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:6_1_1_1_1"/><span class="kw">then </span>
3 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T89" title="INT_1:th.89">INT_1:89</a></span>;<br/><a NAME="E3:6_1_1_1_1"/><span class="kw">then </span>
3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E4:6_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_3.html#T2" target="_self" title="GR_CY_3:th.2">Th2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:6_1_1_1_2"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:6_1_1_1_2"/>
2,3 <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T44" title="INT_2:th.44">INT_2:44</a>, <a class="ref" href="int_2.html#T47" title="INT_2:th.47">INT_2:47</a>, <a class="ref" href="pepin.html#T44" title="PEPIN:th.44">PEPIN:44</a></span>;<br/><a NAME="E3:6_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A6</font></span>: 
2 <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/><a NAME="E4:6_1_1_1_2"/>
3 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:6_1_1_1_2"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="pepin.html#T8" title="PEPIN:th.8">PEPIN:8</a></span>;<br/><a NAME="E5:6_1_1_1_2"/><span class="kw">then </span>
3 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_1_1_1_2"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="wsierp_1.html#T36" title="WSIERP_1:th.36">WSIERP_1:36</a></span>;<br/><a NAME="E6:6_1_1_1_2"/><span class="kw">then </span>
3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:6_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:6_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:6"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#T9" title="INT_3:th.9">INT_3:9</a></span>;<br/>
<a NAME="E6:6"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:6"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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