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

<a NAME="E1:98_1_1"/><i><font color="Green" title="E57">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b> ex <font color="Olive" title="b1">m</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>  st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Olive" title="b1">m</font></span>)</span> &amp; <font color="Olive" title="b1">m</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> );<br/>
<a NAME="E2:98_1_1"/>
 <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 &amp; <font color="Olive" title="b1">k</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="newton.html#E103"><i><font color="Green" title="E56">Lm1</font></i></a></i>;<br/>
<a NAME="E3:98_1_1"/><b>then </b><i><font color="Green" title="E58">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a>, <a class="ref" href="int_2.html#T44" title="INT_2:th.44">INT_2:44</a></i>;<br/>
<a NAME="E4:98_1_1"/><i><font color="Green" title="E59">A3</font></i>: 
for <font color="Olive" title="b1">n</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">n</font> be    <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> ;<br/>

<b>given </b><font color="Maroon" title="c3">m</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> <b> such that </b><a NAME="E1:98_1_1_1"/><i><font color="Green" title="E60">A4</font></i>: 
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">m</font></span>)</span> &amp; <font color="Maroon" title="c3">m</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</font> );<br/>
<a NAME="E2:98_1_1_1"/><i><font color="Green" title="E61">A5</font></i>: 
 ex <font color="Olive" title="b1">k</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>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:98_1_1_1_1"/>
 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">m</font> )
 
<b>by </b><i><a class="ref" href="newton.html#T86" target="_self" title="NEWTON:th.86">Th86</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:98_1_1_1_1"/><i><font color="Green" title="E62">A6</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">k</font>]
 ;<br/>
<a NAME="E4:98_1_1_1_1"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E5:98_1_1_1_1"/>
 ex <font color="Olive" title="b1">k</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>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>by </b><i><a class="txt" href="newton.html#E3:98_1_1_1_1"><i><font color="Green" title="E62">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">k</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> <b> such that </b><br/><a NAME="E4:98_1_1_1"/><i><font color="Green" title="E62">A7</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">k</font>] &amp; ( for <font color="Olive" title="b1">l</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>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">l</font>] holds <br/><font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="newton.html#E2:98_1_1_1"><i><font color="Green" title="E61">A5</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c4">k</font>
;<br/>

<a NAME="E5:98_1_1_1"/><i><font color="Green" title="E63">A8</font></i>: 
 <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:98_1_1_1_2"/>
 <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c4">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:98_1_1_1_2_1"/><i><font color="Green" title="E64">A9</font></i>: 
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c4">k</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c5">s</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>  ;<br/>
<a NAME="E3:98_1_1_1_2_1"/><i><font color="Green" title="E65">A10</font></i>: 
( <font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c6">s</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> )
 
<b>by </b><i><a class="txt" href="newton.html#E1:98_1_1_1_2_1"><i><font color="Green" title="E64">A9</font></i></a>, <a class="ref" href="newton.html#D7" target="_self" title="NEWTON:def.7">Def7</a></i>;<br/>
<a NAME="E4:98_1_1_1_2_1"/><i><font color="Green" title="E66">A11</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="newton.html#E4:98_1_1_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="newton.html#E3:98_1_1_1_2_1"><i><font color="Green" title="E65">A10</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_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="E1:98_1_1_1_2_1_1"/>
( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> or <font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> )
 </span><b>by </b><i><a class="txt" href="newton.html#E4:98_1_1_1_2_1"><i><font color="Green" title="E66">A11</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:98_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 ;<br/><div class="add"><a NAME="E2:98_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:98_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:98_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c6">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 ;<br/><div class="add"><a NAME="E2:98_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E67">A12</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="txt" href="newton.html#E3:98_1_1_1_2_1"><i><font color="Green" title="E65">A10</font></i></a>, <a class="ref" href="newton.html#D7" target="_self" title="NEWTON:def.7">Def7</a></i>;<br/><a NAME="E3:98_1_1_1_2_1_1_2"/>
 <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:98_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="newton.html#E2:98_1_1_1_2_1_1_2"><i><font color="Green" title="E67">A12</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E2:98_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="newton.html#E1:98_1_1_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="newton.html#E4:98_1_1_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="newton.html#T94" target="_self" title="NEWTON:th.94">Th94</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:98_1_1_1"/>
not <font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="ref" href="newton.html#D7" target="_self" title="NEWTON:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E7:98_1_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c4">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="newton.html#E1:98_1_1_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="newton.html#E5:98_1_1_1"><i><font color="Green" title="E63">A8</font></i></a>, <a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></i>;<br/>

<b>thus </b><a NAME="E8:98_1_1_1"/>
<font color="Maroon" title="c4">k</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>
 <b>by </b><i><a class="txt" href="newton.html#E4:98_1_1_1"><i><font color="Green" title="E62">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:98_1_1"/>
for <font color="Olive" title="b1">n</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="newton.html#E3:98_1_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="newton.html#E4:98_1_1"><i><font color="Green" title="E59">A3</font></i></a>);<br/></i>
<a NAME="E6:98_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">m</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>  st <br/>( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Olive" title="b1">m</font></span>)</span> &amp; <font color="Olive" title="b1">m</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> )
 
<b>by </b><i><a class="txt" href="newton.html#E1:98_1_1"><i><font color="Green" title="E57">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:98_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> st <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>


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