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<span class="kw">let </span><font color="Maroon" title="c1">l</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">l</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">two</font> = 2 as   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T28" title="INT_2:th.28">INT_2:28</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">l</font> = <font color="Maroon" title="c1">l</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E3:88"/>
( ( <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  ex <font color="Olive" title="b1">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> ) ) or ( <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b1">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> ) ) or ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">l</font> &amp;  ex <font color="Olive" title="b1">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_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="E1:88_1_1"/>
( <font color="Maroon" title="c3">l</font> <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="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">l</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_1_1_1"><span class="kw">case </span></a><a NAME="E1:88_1_1_1"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c3">l</font> <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">  ex <font color="Olive" title="b1">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c2">two</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">two</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Maroon" title="c2">two</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> )</span><br/><span class="kw">thus </span><a NAME="E2:88_1_1_1"/>
( <font color="Maroon" title="c2">two</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Maroon" title="c2">two</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_1_1_1"><span class="lab"><font color="Green" title="E48">A1</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="88_1_1_2"><span class="kw">case </span></a><a NAME="E1:88_1_1_2"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<font color="Maroon" title="c3">l</font> <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">  ex <font color="Olive" title="b1">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c2">two</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">two</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Maroon" title="c2">two</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> )</span><br/><span class="kw">thus </span><a NAME="E2:88_1_1_2"/>
( <font color="Maroon" title="c2">two</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Maroon" title="c2">two</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_1_1_2"><span class="lab"><font color="Green" title="E48">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="88_1_1_3"><span class="kw">case </span></a><a NAME="E1:88_1_1_3"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> )</span><br/><div class="add"><a NAME="E2:88_1_1_3"/>
<font color="Maroon" title="c3">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">l</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T38" target="_self" title="NEWTON:th.38">Th50</a></span>;<br/><a NAME="E3:88_1_1_3"/><span class="kw">then </span>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">l</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88_1_1_3"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:88_1_1_3"/><span class="kw">then </span>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></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="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:88_1_1_3"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<font color="Maroon" title="c4">j</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E7:88_1_1_3"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<font color="Maroon" title="c4">j</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="newton.html#K11" title="NEWTON:func.11">!</a></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="ref" href="int_2.html#T31" title="INT_2:th.31">INT_2:31</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">j9</font> = <font color="Maroon" title="c4">j</font> as   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:88_1_1_3"><span class="lab"><font color="Green" title="E49">A4</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">j9</font>
; <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="c5">j9</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Maroon" title="c5">j9</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> )</span><br/><a NAME="E9:88_1_1_3"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:88_1_1_3"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/><a NAME="E10:88_1_1_3"/>
<font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:88_1_1_3"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <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="E11:88_1_1_3"/>
( <font color="Maroon" title="c5">j9</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Maroon" title="c5">j9</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">l</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:88_1_1_3"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E9:88_1_1_3"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="newton.html#T39" target="_self" title="NEWTON:th.39">Th51</a>, <a class="ref" href="newton.html#T41" target="_self" title="NEWTON:th.41">Th54</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="E4:88"/>
 ex <font color="Olive" title="b1">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">l</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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