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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> 6 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )</span><br/>

<span class="kw">assume </span><a NAME="E1:8"/><span class="lab"><font color="Green" title="E7">A1</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> 6 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5</span><br/>

<span class="lab"><font color="Green" title="E8">A2</font></span>: <span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 6 = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 3</span>)</span>

<br/>.= 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</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="c1">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#T20" title="INT_4:th.20">INT_4:20</a></span>
<br/>.= 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#T15" title="INT_3:th.15">INT_3:15</a></span>
<br/>.= 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</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> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>
<br/>.= 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</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> 2</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="gr_cy_3.html#T4" target="_self" title="GR_CY_3:th.4">Th4</a></span>
<br/>.= 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3</span>)</span>

<br/>.= 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 3</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#T8" title="INT_3:th.8">INT_3:8</a></span>
<br/>.= 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T5" title="PEPIN:th.5">PEPIN:5</a></span>
;<br/>
<span class="lab"><font color="Green" title="E9">A3</font></span>: <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 6 = 
<span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span>

<br/>.= 
3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#T20" title="INT_4:th.20">INT_4:20</a></span>
<br/>.= 
3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_3.html#T3" target="_self" title="GR_CY_3:th.3">Th3</a></span>
;<br/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 6 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">6 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 6
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_3.html#T8" title="INT_3:th.8">INT_3:8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 6

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 6
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="int_3.html#T14" title="INT_3:th.14">INT_3:14</a></span>
<br/>.= 
5
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:8"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 6 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ; <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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