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

<span class="kw">set </span><font color="Maroon" title="c1">x</font> = <span class="p1">(<span class="default">5 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4</span>)</span>;<br/>
<span class="kw">thus </span><a NAME="E1:58"/>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">5 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="number02.html#E61"><span class="lab"><font color="Green" title="E44">Lm6</font></span></a>, <a class="txt" href="number02.html#E62"><span class="lab"><font color="Green" title="E45">Lm7</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="number02.html#D1" target="_self" onmouseover="rs('number02/D1')" onmouseout="rh()">NUMBER02:def 1</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"> (  not <span class="p1">(<span class="default">5 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4</span>)</span> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <br/>(  not <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 5 or  not <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4</span>)</span> is  <a href="number02.html#V1" title="NUMBER02:attr.1">composite</a>  ) ) )</span><br/>

<a NAME="E2:58"/>
<span class="p1">(<span class="default">5 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 17 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 113
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="number02.html#E61"><span class="lab"><font color="Green" title="E44">Lm6</font></span></a>, <a class="txt" href="number02.html#E62"><span class="lab"><font color="Green" title="E45">Lm7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:58"/>
 not <span class="p1">(<span class="default">5 <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4</span>)</span> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number02.html#T13" target="_self" onmouseover="rs('number02/T13')" onmouseout="rh()">Th13</a>, <a class="ref" href="pepin.html#T60" onmouseover="rs('pepin/T60')" onmouseout="rh()">PEPIN:60</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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <br/>(  not <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 5 or  not <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4</span>)</span> is  <a href="number02.html#V1" title="NUMBER02:attr.1">composite</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><a NAME="E5:58"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 5
 <span class="kw">and </span><br/><a NAME="E6:58"/><span class="lab"><font color="Green" title="E51">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><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> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> 4</span>)</span> is  <a href="number02.html#V1" title="NUMBER02:attr.1">composite</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"> contradiction</span><br/>

<a NAME="E7:58"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 5 <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="#E5:58"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="ref" href="int_1.html#T52" onmouseover="rs('int_1/T52')" onmouseout="rh()">INT_1:52</a></span>;<br/>
<a NAME="E8:58"/><span class="kw">then </span>
 not  not <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  ...  &amp;  not <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 
;<br/>
<span class="kw">hence </span><a NAME="E9:58"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="number02.html#T56" target="_self" onmouseover="rs('number02/T56')" onmouseout="rh()">Th56</a>, <a class="ref" href="number02.html#T57" target="_self" onmouseover="rs('number02/T57')" onmouseout="rh()">Th57</a>, <a class="ref" href="number02.html#T58" target="_self" onmouseover="rs('number02/T58')" onmouseout="rh()">Th58</a>, <a class="ref" href="number02.html#T59" target="_self" onmouseover="rs('number02/T59')" onmouseout="rh()">Th59</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>
