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

<a NAME="E1:101_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b> ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Olive">m</font></span>)</span> &amp; <font color="Olive">m</font> is   <a href="newton.html#NM1">Prime</a> );<br/>
<a NAME="E2:101_1_1"/>
 <a href="newton.html#K14" target="_self">SetPrimenumber</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 &amp; <font color="Olive">k</font> is <a href="int_2.html#V1">prime</a> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:101_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="int_2.html#T44">INT_2:44</a></i>;<br/>
<a NAME="E4:101_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>given </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:101_1_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">m</font> is   <a href="newton.html#NM1">Prime</a> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> );<br/>
<a NAME="E2:101_1_1_1"/><i><font color="Green">E47</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:101_1_1_1_1"/>
 ex <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">p</font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="newton.html#T7" target="_self">Th7</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:101_1_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/>
<a NAME="E4:101_1_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E5:101_1_1_1_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:101_1_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>] &amp; ( for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">l</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="ref" href="newton.html#T11" target="_self">Th11</a>);<br/></i>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<a NAME="E5:101_1_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">m</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:101_1_1_1_2"/>
 <a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">m</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:101_1_1_1_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">k</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:101_1_1_1_2_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">s</font> is <a href="int_2.html#V1">prime</a> )
 
<b>by </b><i><a class="ref" href="newton.html#T16" target="_self">Th16</a>, <a class="txt" href="newton.html#E4:101_1_1_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E4:101_1_1_1_2_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="newton.html#T18" target="_self">Th18</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_1_1_1_2_1_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> or <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 </span><b>by </b><i><a class="ref" href="newton.html#T19" target="_self">Th19</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:101_1_1_1_2_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:101_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:101_1_1_1_2_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">m</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:101_1_1_1_2_1_1_2"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:101_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="newton.html#T18" target="_self">Th18</a>, <a class="txt" href="newton.html#E4:101_1_1_1"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E3:101_1_1_1_2_1_1_2"/>
 <a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">m</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">m</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:101_1_1_1_2_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">m</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="newton.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E2:101_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">m</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:101_1_1_1"/>
not <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="newton.html#E4:101_1_1_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:101_1_1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a>, <a class="ref" href="newton.html#T15" target="_self">Th15</a>, <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/>

<b>thus </b><a NAME="E8:101_1_1_1"/>
<font color="Maroon">k</font> is   <a href="newton.html#NM1">Prime</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:101_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="ref" href="newton.html#T9" target="_self">Th9</a>);<br/></i>
<a NAME="E6:101_1_1"/><b>then </b>
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Olive">m</font></span>)</span> &amp; <font color="Olive">m</font> is   <a href="newton.html#NM1">Prime</a> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:101_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>


</div>
