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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = 2 as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="ref" href="int_2.html#T44">INT_2:44</a></i>;<br/>
<a NAME="E2:95"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> ) ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> ) ) or ( 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:95_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 or 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="newton.html#E93"><i><font color="Green">Lemma51</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_1"><b>case </b></a><a NAME="E1:95_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/><b>thus </b><a NAME="E2:95_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="newton.html#E1:95_1_1_1"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_2"><b>case </b></a><a NAME="E1:95_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/><b>thus </b><a NAME="E2:95_1_1_2"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="newton.html#E1:95_1_1_2"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="95_1_1_3"><b>case </b></a><a NAME="E1:95_1_1_3"/><i><font color="Green">E53</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:95_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K12" target="_self">!</a> 
 <b>by </b><i><a class="ref" href="newton.html#T50" target="_self">Th50</a></i>;<br/><a NAME="E3:95_1_1_3"/><b>then </b>
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K12" target="_self">!</a> 
 <b>by </b><i><a class="txt" href="newton.html#E1:95_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:95_1_1_3"/><b>then </b>
2 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K12" target="_self">!</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:95_1_1_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="int_2.html#V1">prime</a>
 <b>and </b><br/><a NAME="E7:95_1_1_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K12" target="_self">!</a> </span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#T48">INT_2:48</a></i>;<br/><a NAME="E8:95_1_1_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="newton.html#E6:95_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E9:95_1_1_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="newton.html#E6:95_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="txt" href="newton.html#E6:95_1_1_3"><i><font color="Green">E54</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/><b>thus </b><a NAME="E11:95_1_1_3"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="newton.html#E7:95_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="newton.html#E8:95_1_1_3"><i><font color="Green">E56</font></i></a>, <a class="txt" href="newton.html#E9:95_1_1_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="newton.html#T55" target="_self">Th55</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:95"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>


</div>
