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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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> 2 implies <span class="p1">(<span class="default"><a href="gr_cy_3.html#K1" title="GR_CY_3:func.1">Mersenne</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 )</span><br/>

<a NAME="E1:32"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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="E2:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <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="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:32"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="gr_cy_3.html#K1" title="GR_CY_3:func.1">Mersenne</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7</span><br/>

<a NAME="E4:32"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E5:32"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T14" title="XREAL_1:th.14">XREAL_1:14</a></span>;<br/>
<a NAME="E6:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E7:32"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/>
<span class="p1">(<span class="default"><a href="gr_cy_3.html#K1" title="GR_CY_3:func.1">Mersenne</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 = 
<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">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 8
<span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T26" title="PEPIN:th.26">PEPIN:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 8
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_6.html#T7" title="INT_6:th.7">INT_6:7</a></span>
<br/>.= 
<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"><span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 8
<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/>.= 
<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"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 8
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="pepin.html#T26" title="PEPIN:th.26">PEPIN:26</a></span>
<br/>.= 
<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"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 8
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T45" title="NAT_D:th.45">NAT_D:45</a></span>
<br/>.= 
<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">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 8
<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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 2</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 8
<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="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 8
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="pepin.html#T36" title="PEPIN:th.36">PEPIN:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default">8 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> 8
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T61" title="NAT_D:th.61">NAT_D:61</a></span>
<br/>.= 
7
<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="E9:32"/>
<span class="p1">(<span class="default"><a href="gr_cy_3.html#K1" title="GR_CY_3:func.1">Mersenne</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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