<?xml version="1.0"?>
<div class="add">

<a NAME="E1:10"/>
8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 4
 
;<br/>
<a NAME="E2:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A1</font></span>: 
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 8
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/>
<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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 5 or <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c1">p</font> <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/>

<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> 8;<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="E4:10"/><span class="lab"><font color="Green" title="E8">A2</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="E5:10"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 5
 ; <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="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c1">p</font> <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/>

<div><span class="lab"><font color="Green" title="E10">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:10_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
( <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <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> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 or <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_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="E1:10_1_1_1"/>
( <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <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> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 or <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:10_1_1_1_1"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:10_1_1_1_1"/><span class="kw">then </span>
8 <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#T62" title="INT_1:th.62">INT_1:62</a></span>;<br/><a NAME="E3:10_1_1_1_1"/><span class="kw">then </span>
2 <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="txt" href="#E2:10"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="int_2.html#T9" title="INT_2:th.9">INT_2:9</a></span>;<br/><a NAME="E4:10_1_1_1_1"/><span class="kw">then </span>
2 <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#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:10_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="firebrick">::  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="10_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:10_1_1_1_2"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:10_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 8</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 8</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> <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="int_1.html#T59" title="INT_1:th.59">INT_1:59</a></span>;<br/><a NAME="E3:10_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 8</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 4
 ;<br/><a NAME="E4:10_1_1_1_2"/><span class="kw">then </span>
4 <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="ref" href="int_1.html#D3" title="INT_1:def.3">INT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:10_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T29" title="INT_2:th.29">INT_2:29</a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:10_1_1_1_3"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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/><div class="add"><span class="kw">then </span><font color="Maroon" title="c1">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 8</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 8</span>)</span> <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="int_1.html#T59" title="INT_1:th.59">INT_1:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 8</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2

;<br/><a NAME="E3:10_1_1_1_3"/><span class="kw">then </span>
2 <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#D3" title="INT_1:def.3">INT_1:def 3</a></span>;<br/><a NAME="E4:10_1_1_1_3"/><span class="kw">then </span>
2 <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#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:10_1_1_1_3"/>
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="firebrick">::  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="10_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:10_1_1_1_4"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><font color="Maroon" title="c1">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 8</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 8</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 4
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T59" title="INT_1:th.59">INT_1:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 8</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2

;<br/><a NAME="E3:10_1_1_1_4"/><span class="kw">then </span>
2 <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#D3" title="INT_1:def.3">INT_1:def 3</a></span>;<br/><a NAME="E4:10_1_1_1_4"/><span class="kw">then </span>
2 <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#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:10_1_1_1_4"/>
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="firebrick">::  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="10_1_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:10_1_1_1_5"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:10_1_1_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 8</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 8</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="int_1.html#T59" title="INT_1:th.59">INT_1:59</a></span>;<br/><a NAME="E3:10_1_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 8</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2
 ;<br/><a NAME="E4:10_1_1_1_5"/><span class="kw">then </span>
2 <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="ref" href="int_1.html#D3" title="INT_1:def.3">INT_1:def 3</a></span>;<br/><a NAME="E5:10_1_1_1_5"/><span class="kw">then </span>
2 <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#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E6:10_1_1_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E8">A2</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_6"><span class="kw">suppose </span></a><a NAME="E1:10_1_1_1_6"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><font color="Maroon" title="c1">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 8</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 8</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 6
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T59" title="INT_1:th.59">INT_1:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 8</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2

;<br/><a NAME="E3:10_1_1_1_6"/><span class="kw">then </span>
2 <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#D3" title="INT_1:def.3">INT_1:def 3</a></span>;<br/><a NAME="E4:10_1_1_1_6"/><span class="kw">then </span>
2 <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#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:10_1_1_1_6"/>
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="firebrick">::  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="E3:10_1"/>
contradiction
 ; <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>
<a NAME="E7:10"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 7 <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_d.html#T62" title="NAT_D:th.62">NAT_D:62</a></span>;<br/>
<a NAME="E8:10"/><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> 8 <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> 7
 
<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="E9:10"/>
( <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 8 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c1">p</font> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="nat_1.html#T31" title="NAT_1:th.31">NAT_1:31</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>
