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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> is  <a href="number08.html#V1" title="NUMBER08:attr.1">having_exactly_one_prime_divisor</a>  implies  ex <font color="Olive" title="b1">k</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> st <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="Olive" title="b1">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:132"/><span class="lab"><font color="Green" title="E103">A1</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> 
 ; <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">k</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> st <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="Olive" title="b1">k</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E3:132"/><span class="lab"><font color="Green" title="E104">A2</font></span>: 
<font color="Maroon" title="c3">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">and </span><br/><a NAME="E4:132"/><span class="lab"><font color="Green" title="E105">A3</font></span>: 
 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="c3">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>
 ;<br/>
<a NAME="E5:132"/><span class="lab"><font color="Green" title="E106">A4</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> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:132"><span class="lab"><font color="Green" title="E103">A1</font></span></a>, <a class="txt" href="#E3:132"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="ref" href="number08.html#D7" target="_self" onmouseover="rs('number08/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="132_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 <!--HUMANRECFAILED--> for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">k</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:132_1"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  holds <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">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/><a NAME="E2:132_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s</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>  st <br/>( <font color="Olive" title="b1">s</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Olive" title="b1">s</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groupp_1.html#T1" onmouseover="rs('groupp_1/T1')" onmouseout="rh()">GROUPP_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:132_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:132"><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>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E8:132"/><span class="lab"><font color="Green" title="E107">A5</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">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:132"><span class="lab"><font color="Green" title="E106">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c4">k</font> as   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">by</span> <span class="lab"><a class="txt" href="#E1:132"><span class="lab"><font color="Green" title="E103">A1</font></span></a>, <a class="txt" href="#E8:132"><span class="lab"><font color="Green" title="E107">A5</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">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"> <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="c5">k</font></span><br/>

<span class="kw">thus </span><a NAME="E10:132"/>
<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="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:132"><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/>


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