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<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 7 implies  ex <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Olive" title="b2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">q</font> &amp; ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> or <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 ) &amp; <font color="Olive" title="b2">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b3">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Olive" title="b1">m</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:133"/><span class="lab"><font color="Green" title="E103">A1</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 7
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Olive" title="b2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">q</font> &amp; ( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> or <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 ) &amp; <font color="Olive" title="b2">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b3">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Olive" title="b1">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:133"/><span class="lab"><font color="Green" title="E104">A2</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  holds <br/>(  not <font color="Olive" title="b2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">q</font> or (  not <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> &amp;  not <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  not <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 ) or  not <font color="Olive" title="b2">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Olive" title="b1">m</font> or  not <font color="Olive" title="b3">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Olive" title="b1">m</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:133"/><span class="kw">then </span>
(  not 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> or  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a>, <a class="ref" href="xprimes1.html#T3" onmouseover="rs('xprimes1/T3')" onmouseout="rh()">XPRIMES1:3</a></span>;<br/>
<a NAME="E4:133"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A3</font></span>: 
 not 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number04.html#T3" onmouseover="rs('number04/T3')" onmouseout="rh()">NUMBER04:3</a></span>;<br/>
<a NAME="E5:133"/>
(  not 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a>, <a class="ref" href="xprimes1.html#T3" onmouseover="rs('xprimes1/T3')" onmouseout="rh()">XPRIMES1:3</a></span>;<br/>
<a NAME="E6:133"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A4</font></span>: 
 not 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number04.html#T3" onmouseover="rs('number04/T3')" onmouseout="rh()">NUMBER04:3</a></span>;<br/>
<a NAME="E7:133"/>
(  not 2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 or  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a>, <a class="ref" href="xprimes1.html#T3" onmouseover="rs('xprimes1/T3')" onmouseout="rh()">XPRIMES1:3</a></span>;<br/>
<a NAME="E8:133"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A5</font></span>: 
 not 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number04.html#T3" onmouseover="rs('number04/T3')" onmouseout="rh()">NUMBER04:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:133"/><span class="lab"><font color="Green" title="E108">A6</font></span>: 
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font> or <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 or <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 or <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 or <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 5 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number02.html#T26" onmouseover="rs('number02/T26')" onmouseout="rh()">NUMBER02:26</a></span>;<br/>
<a NAME="E11:133"/><span class="lab"><font color="Green" title="E109">A7</font></span>: 
<font color="Maroon" title="c2">n</font> is  <a href="number08.html#V1" title="NUMBER08:attr.1">having_exactly_one_prime_divisor</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="133_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E2:133_1"/><span class="lab"><font color="Green" title="E110">A8</font></span>: 
<font color="Maroon" title="c4">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E3:133_1"/><span class="lab"><font color="Green" title="E111">A9</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E103">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a>, <a class="ref" href="int_2.html#T31" onmouseover="rs('int_2/T31')" onmouseout="rh()">INT_2:31</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">p</font> as   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133_1"><span class="lab"><font color="Green" title="E110">A8</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">p</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="number08.html#D6" target="_self" onmouseover="rs('number08/D6')" onmouseout="rh()">NUMBER08:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> &amp; (  for <font color="Olive" title="b1">r</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font> holds <br/> not <font color="Olive" title="b1">r</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:133_1"/>
( <font color="Maroon" title="c5">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> &amp; (  for <font color="Olive" title="b1">r</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font> holds <br/> not <font color="Olive" title="b1">r</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E3:133_1"><span class="lab"><font color="Green" title="E111">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E12:133"/><span class="lab"><font color="Green" title="E110">A10</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is  <a href="number08.html#V1" title="NUMBER08:attr.1">having_exactly_one_prime_divisor</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="133_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:133_2"/>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<a NAME="E2:133_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 7
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E103">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:133_2"/><span class="lab"><font color="Green" title="E111">A11</font></span>: 
<font color="Maroon" title="c4">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E5:133_2"/><span class="lab"><font color="Green" title="E112">A12</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T31" onmouseover="rs('int_2/T31')" onmouseout="rh()">INT_2:31</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">p</font> as   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:133_2"><span class="lab"><font color="Green" title="E111">A11</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">p</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="number08.html#D6" target="_self" onmouseover="rs('number08/D6')" onmouseout="rh()">NUMBER08:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; (  for <font color="Olive" title="b1">r</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font> holds <br/> not <font color="Olive" title="b1">r</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:133_2"/>
( <font color="Maroon" title="c5">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; (  for <font color="Olive" title="b1">r</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font> holds <br/> not <font color="Olive" title="b1">r</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E5:133_2"><span class="lab"><font color="Green" title="E112">A12</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E13:133"/><span class="lab"><font color="Green" title="E111">A13</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 is  <a href="number08.html#V1" title="NUMBER08:attr.1">having_exactly_one_prime_divisor</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="133_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:133_3"/>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<a NAME="E2:133_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 7
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E103">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:133_3"/><span class="lab"><font color="Green" title="E112">A14</font></span>: 
<font color="Maroon" title="c4">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E5:133_3"/><span class="lab"><font color="Green" title="E113">A15</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T31" onmouseover="rs('int_2/T31')" onmouseout="rh()">INT_2:31</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">p</font> as   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:133_3"><span class="lab"><font color="Green" title="E112">A14</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">p</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="number08.html#D6" target="_self" onmouseover="rs('number08/D6')" onmouseout="rh()">NUMBER08:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 &amp; (  for <font color="Olive" title="b1">r</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font> holds <br/> not <font color="Olive" title="b1">r</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:133_3"/>
( <font color="Maroon" title="c5">p</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 &amp; (  for <font color="Olive" title="b1">r</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p</font> holds <br/> not <font color="Olive" title="b1">r</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E5:133_3"><span class="lab"><font color="Green" title="E113">A15</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="133_4"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:133_4"/>
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font> or <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 or <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 or <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4 or <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 5 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:133"><span class="lab"><font color="Green" title="E108">A6</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="133_4_1"><span class="kw">suppose </span></a><a NAME="E1:133_4_1"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">hence </span><a NAME="E2:133_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:133"><span class="lab"><font color="Green" title="E105">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="133_4_2"><span class="kw">suppose </span></a><a NAME="E1:133_4_2"/><span class="lab"><font color="Green" title="E112">A16</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">consider </span><font color="Maroon" title="c4">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:133_4_2"/><span class="lab"><font color="Green" title="E113">A17</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="number08.html#K4" title="NUMBER08:func.4">the_only_divisor_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:133"><span class="lab"><font color="Green" title="E110">A10</font></span></a>, <a class="ref" href="number08.html#T78" target="_self" onmouseover="rs('number08/T78')" onmouseout="rh()">Th78</a></span>;<br/><a NAME="E4:133_4_2"/>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_4_2"><span class="lab"><font color="Green" title="E112">A16</font></span></a></span>;<br/><a NAME="E5:133_4_2"/><span class="kw">then </span>
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E6:133_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A18</font></span>: 
 <a href="number08.html#K4" title="NUMBER08:func.4">the_only_divisor_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:133"><span class="lab"><font color="Green" title="E110">A10</font></span></a>, <a class="ref" href="number08.html#D7" target="_self" onmouseover="rs('number08/D7')" onmouseout="rh()">Def7</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">s</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:133_4_2"/><span class="lab"><font color="Green" title="E115">A19</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="number08.html#K4" title="NUMBER08:func.4">the_only_divisor_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c5">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:133"><span class="lab"><font color="Green" title="E111">A13</font></span></a>, <a class="ref" href="number08.html#T78" target="_self" onmouseover="rs('number08/T78')" onmouseout="rh()">Th78</a></span>;<br/><a NAME="E9:133_4_2"/>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_4_2"><span class="lab"><font color="Green" title="E112">A16</font></span></a></span>;<br/><a NAME="E10:133_4_2"/><span class="kw">then </span>
3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ;<br/><a NAME="E11:133_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A20</font></span>: 
 <a href="number08.html#K4" title="NUMBER08:func.4">the_only_divisor_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:133"><span class="lab"><font color="Green" title="E111">A13</font></span></a>, <a class="ref" href="number08.html#D7" target="_self" onmouseover="rs('number08/D7')" onmouseout="rh()">Def7</a>, <a class="ref" href="xprimes1.html#T3" onmouseover="rs('xprimes1/T3')" onmouseout="rh()">XPRIMES1:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E117">A21</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="133_4_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:133_4_2_1"/>
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:133_4_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 7 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:133_4_2"><span class="lab"><font color="Green" title="E113">A17</font></span></a>, <a class="txt" href="#E6:133_4_2"><span class="lab"><font color="Green" title="E114">A18</font></span></a>, <a class="txt" href="number08.html#E12"><span class="lab"><font color="Green" title="E12">Lm12</font></span></a>, <a class="ref" href="prepower.html#T93" onmouseover="rs('prepower/T93')" onmouseout="rh()">PREPOWER:93</a></span>;<br/><span class="kw">hence </span><a NAME="E3:133_4_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E103">A1</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E13:133_4_2"/>
<span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:133_4_2"><span class="lab"><font color="Green" title="E113">A17</font></span></a>, <a class="txt" href="#E8:133_4_2"><span class="lab"><font color="Green" title="E115">A19</font></span></a>, <a class="txt" href="#E6:133_4_2"><span class="lab"><font color="Green" title="E114">A18</font></span></a>, <a class="txt" href="#E11:133_4_2"><span class="lab"><font color="Green" title="E116">A20</font></span></a></span>;<br/><a NAME="E14:133_4_2"/><span class="kw">then </span>
( ( <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T73" target="_self" onmouseover="rs('number08/T73')" onmouseout="rh()">Th73</a></span>;<br/><span class="kw">hence </span><a NAME="E15:133_4_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:133_4_2"><span class="lab"><font color="Green" title="E117">A21</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="133_4_3"><span class="kw">suppose </span></a><a NAME="E1:133_4_3"/><span class="lab"><font color="Green" title="E112">A22</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">consider </span><font color="Maroon" title="c4">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:133_4_3"/><span class="lab"><font color="Green" title="E113">A23</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="number08.html#K4" title="NUMBER08:func.4">the_only_divisor_of</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:133"><span class="lab"><font color="Green" title="E109">A7</font></span></a>, <a class="ref" href="number08.html#T78" target="_self" onmouseover="rs('number08/T78')" onmouseout="rh()">Th78</a></span>;<br/><a NAME="E4:133_4_3"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_4_3"><span class="lab"><font color="Green" title="E112">A22</font></span></a></span>;<br/><a NAME="E5:133_4_3"/><span class="kw">then </span>
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 ;<br/><a NAME="E6:133_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A24</font></span>: 
 <a href="number08.html#K4" title="NUMBER08:func.4">the_only_divisor_of</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:133"><span class="lab"><font color="Green" title="E109">A7</font></span></a>, <a class="ref" href="number08.html#D7" target="_self" onmouseover="rs('number08/D7')" onmouseout="rh()">Def7</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">s</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:133_4_3"/><span class="lab"><font color="Green" title="E115">A25</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="number08.html#K4" title="NUMBER08:func.4">the_only_divisor_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c5">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:133"><span class="lab"><font color="Green" title="E110">A10</font></span></a>, <a class="ref" href="number08.html#T78" target="_self" onmouseover="rs('number08/T78')" onmouseout="rh()">Th78</a></span>;<br/><a NAME="E9:133_4_3"/>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_4_3"><span class="lab"><font color="Green" title="E112">A22</font></span></a></span>;<br/><a NAME="E10:133_4_3"/><span class="kw">then </span>
3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E11:133_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A26</font></span>: 
 <a href="number08.html#K4" title="NUMBER08:func.4">the_only_divisor_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:133"><span class="lab"><font color="Green" title="E110">A10</font></span></a>, <a class="ref" href="number08.html#D7" target="_self" onmouseover="rs('number08/D7')" onmouseout="rh()">Def7</a>, <a class="ref" href="xprimes1.html#T3" onmouseover="rs('xprimes1/T3')" onmouseout="rh()">XPRIMES1:3</a></span>;<br/><div><span class="lab"><font color="Green" title="E117">A27</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="133_4_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:133_4_3_1"/>
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:133_4_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A28</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 7 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:133_4_3"><span class="lab"><font color="Green" title="E113">A23</font></span></a>, <a class="txt" href="#E6:133_4_3"><span class="lab"><font color="Green" title="E114">A24</font></span></a>, <a class="txt" href="number08.html#E12"><span class="lab"><font color="Green" title="E12">Lm12</font></span></a>, <a class="ref" href="prepower.html#T93" onmouseover="rs('prepower/T93')" onmouseout="rh()">PREPOWER:93</a></span>;<br/><a NAME="E3:133_4_3_1"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 7 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E103">A1</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:133_4_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A29</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 8
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:133_4_3_1"><span class="lab"><font color="Green" title="E118">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E5:133_4_3_1"/>
 not 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 5 is  <a href="number08.html#V1" title="NUMBER08:attr.1">having_exactly_one_prime_divisor</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T77" target="_self" onmouseover="rs('number08/T77')" onmouseout="rh()">Th77</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a>, <a class="ref" href="xprimes1.html#T5" onmouseover="rs('xprimes1/T5')" onmouseout="rh()">XPRIMES1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E6:133_4_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:133"><span class="lab"><font color="Green" title="E111">A13</font></span></a>, <a class="txt" href="#E4:133_4_3_1"><span class="lab"><font color="Green" title="E119">A29</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E13:133_4_3"/>
<span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:133_4_3"><span class="lab"><font color="Green" title="E113">A23</font></span></a>, <a class="txt" href="#E8:133_4_3"><span class="lab"><font color="Green" title="E115">A25</font></span></a>, <a class="txt" href="#E6:133_4_3"><span class="lab"><font color="Green" title="E114">A24</font></span></a>, <a class="txt" href="#E11:133_4_3"><span class="lab"><font color="Green" title="E116">A26</font></span></a></span>;<br/><a NAME="E14:133_4_3"/><span class="kw">then </span>
( ( <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T73" target="_self" onmouseover="rs('number08/T73')" onmouseout="rh()">Th73</a></span>;<br/><span class="kw">hence </span><a NAME="E15:133_4_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:133_4_3"><span class="lab"><font color="Green" title="E117">A27</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="133_4_4"><span class="kw">suppose </span></a><a NAME="E1:133_4_4"/><span class="lab"><font color="Green" title="E112">A30</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">consider </span><font color="Maroon" title="c4">s</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:133_4_4"/><span class="lab"><font color="Green" title="E113">A31</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="number08.html#K4" title="NUMBER08:func.4">the_only_divisor_of</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:133"><span class="lab"><font color="Green" title="E109">A7</font></span></a>, <a class="ref" href="number08.html#T78" target="_self" onmouseover="rs('number08/T78')" onmouseout="rh()">Th78</a></span>;<br/><a NAME="E4:133_4_4"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_4_4"><span class="lab"><font color="Green" title="E112">A30</font></span></a></span>;<br/><a NAME="E5:133_4_4"/><span class="kw">then </span>
3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 ;<br/><a NAME="E6:133_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A32</font></span>: 
 <a href="number08.html#K4" title="NUMBER08:func.4">the_only_divisor_of</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:133"><span class="lab"><font color="Green" title="E109">A7</font></span></a>, <a class="ref" href="number08.html#D7" target="_self" onmouseover="rs('number08/D7')" onmouseout="rh()">Def7</a>, <a class="ref" href="xprimes1.html#T3" onmouseover="rs('xprimes1/T3')" onmouseout="rh()">XPRIMES1:3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c5">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:133_4_4"/><span class="lab"><font color="Green" title="E115">A33</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="number08.html#K4" title="NUMBER08:func.4">the_only_divisor_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:133"><span class="lab"><font color="Green" title="E110">A10</font></span></a>, <a class="ref" href="number08.html#T78" target="_self" onmouseover="rs('number08/T78')" onmouseout="rh()">Th78</a></span>;<br/><a NAME="E9:133_4_4"/>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133_4_4"><span class="lab"><font color="Green" title="E112">A30</font></span></a></span>;<br/><a NAME="E10:133_4_4"/><span class="kw">then </span>
2 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E11:133_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A34</font></span>: 
 <a href="number08.html#K4" title="NUMBER08:func.4">the_only_divisor_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:133"><span class="lab"><font color="Green" title="E110">A10</font></span></a>, <a class="ref" href="number08.html#D7" target="_self" onmouseover="rs('number08/D7')" onmouseout="rh()">Def7</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a></span>;<br/><div><span class="lab"><font color="Green" title="E117">A35</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="133_4_4_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:133_4_4_1"/>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:133_4_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 7 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:133_4_4"><span class="lab"><font color="Green" title="E115">A33</font></span></a>, <a class="txt" href="#E11:133_4_4"><span class="lab"><font color="Green" title="E116">A34</font></span></a>, <a class="txt" href="number08.html#E12"><span class="lab"><font color="Green" title="E12">Lm12</font></span></a>, <a class="ref" href="prepower.html#T93" onmouseover="rs('prepower/T93')" onmouseout="rh()">PREPOWER:93</a></span>;<br/><span class="kw">hence </span><a NAME="E3:133_4_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:133"><span class="lab"><font color="Green" title="E103">A1</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E13:133_4_4"/>
<span class="p1">(<span class="default">2 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:133_4_4"><span class="lab"><font color="Green" title="E115">A33</font></span></a>, <a class="txt" href="#E3:133_4_4"><span class="lab"><font color="Green" title="E113">A31</font></span></a>, <a class="txt" href="#E11:133_4_4"><span class="lab"><font color="Green" title="E116">A34</font></span></a>, <a class="txt" href="#E6:133_4_4"><span class="lab"><font color="Green" title="E114">A32</font></span></a></span>;<br/><a NAME="E14:133_4_4"/><span class="kw">then </span>
( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T67" target="_self" onmouseover="rs('number08/T67')" onmouseout="rh()">Th67</a></span>;<br/><span class="kw">hence </span><a NAME="E15:133_4_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:133_4_4"><span class="lab"><font color="Green" title="E117">A35</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="133_4_5"><span class="kw">suppose </span></a><a NAME="E1:133_4_5"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 4
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:133_4_5"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E3:133_4_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:133"><span class="lab"><font color="Green" title="E107">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="133_4_6"><span class="kw">suppose </span></a><a NAME="E1:133_4_6"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">6 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 5
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:133_4_6"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E3:133_4_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:133"><span class="lab"><font color="Green" title="E106">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
