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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">p</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> implies ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">p</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E2:37"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is   <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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">p</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:37_1"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c1">a</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="txt" href="#E1:37"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="gr_cy_3.html#E36"><span class="lab"><font color="Green" title="E18">Lm6</font></span></a></span>;<br/><a NAME="E2:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<font color="Maroon" title="c1">a</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/><a NAME="E3:37_1"/>
<font color="Maroon" title="c2">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="txt" href="#E1:37"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:37_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></span>;<br/><a NAME="E5:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">p</font></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="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:37_1_1"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:37_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <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> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E3:37_1_1"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is    <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/><a NAME="E4:37_1_1"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">p</font></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="#E1:37"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="gr_cy_3.html#E36"><span class="lab"><font color="Green" title="E18">Lm6</font></span></a>, <a class="ref" href="gr_cy_3.html#T27" target="_self" title="GR_CY_3:th.27">Th27</a></span>;<br/><span class="kw">hence </span><a NAME="E5:37_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E5:37_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="ref" href="nat_4.html#T12" title="NAT_4:th.12">NAT_4:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:37_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <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="c2">p</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a></span><br/><span class="kw">assume </span><a NAME="E8:37_1"/>
<font color="Maroon" title="c2">p</font> is  not   <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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">n</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> <span class="kw"> such that </span><br/><a NAME="E10:37_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">and </span><br/><a NAME="E11:37_1"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">and </span><br/><a NAME="E12:37_1"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="nat_4.html#T12" title="NAT_4:th.12">NAT_4:12</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c4">q</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E14:37_1"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:37_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><a NAME="E15:37_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E16:37_1"/><span class="kw">then </span>
2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></span>;<br/><a NAME="E17:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A13</font></span>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c3">n</font>
 <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/><a NAME="E18:37_1"/>
2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="prepower.html#T9" title="PREPOWER:th.9">PREPOWER:9</a></span>;<br/><a NAME="E19:37_1"/><span class="kw">then </span>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:37_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E20:37_1"/><span class="kw">then </span>
2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font>
 <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/><a NAME="E21:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A14</font></span>: 
2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font></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="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E22:37_1"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="gr_cy_3.html#E36"><span class="lab"><font color="Green" title="E18">Lm6</font></span></a></span>;<br/><a NAME="E23:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is    <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a>, <a class="ref" href="prepower.html#T11" title="PREPOWER:th.11">PREPOWER:11</a></span>;<br/><a NAME="E24:37_1"/>
<font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E12:37_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="pepin.html#T66" title="PEPIN:th.66">PEPIN:66</a></span>;<br/><a NAME="E25:37_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">p</font></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="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E26:37_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">q</font></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="#E1:37_1"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E11:37_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="ref" href="gr_cy_3.html#T27" target="_self" title="GR_CY_3:th.27">Th27</a>, <a class="ref" href="pepin.html#T25" title="PEPIN:th.25">PEPIN:25</a></span>;<br/><a NAME="E27:37_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">p</font></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="#E14:37_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>;<br/><span class="kw">hence </span><a NAME="E28:37_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E25:37_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E23:37_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="txt" href="#E21:37_1"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="ref" href="nat_4.html#T12" title="NAT_4:th.12">NAT_4:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:37"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">p</font> is   <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"> verum</span><br/>


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