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

<b>let </b><font color="Maroon">f</font> be   <a href="newton.html#NM1">Prime</a>, <font color="Maroon">g</font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:101_1_2"/><i><font color="Green">E85</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">f</font></span>)</span>
 <b>and </b><br/><a NAME="E2:101_1_2"/><i><font color="Green">E86</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">g</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E3:101_1_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E4:101_1_2"/>
( ( <font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> &amp; <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> &amp; <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_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_2_1_1"/>
( <font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> or <font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> )
 </span><b>by </b><i><a class="ref" href="newton.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_1"><b>case </b></a><a NAME="E1:101_1_2_1_1_1"/>
<font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:101_1_2_1_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">f</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">f</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">g</font>
 <b>by </b><i/>;<br/><a NAME="E3:101_1_2_1_1_1"/><i><font color="Green">E103</font></i>: 
not <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><a NAME="E4:101_1_2_1_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">f</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a>, <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><b>hence </b><a NAME="E5:101_1_2_1_1_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="newton.html#T11" target="_self">Th11</a>, <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2_1_1_2"><b>case </b></a><a NAME="E1:101_1_2_1_1_2"/>
<font color="Maroon">g</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:101_1_2_1_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">g</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">g</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">f</font>
 <b>by </b><i/>;<br/><a NAME="E3:101_1_2_1_1_2"/><i><font color="Green">E105</font></i>: 
not <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/><a NAME="E4:101_1_2_1_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">g</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</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">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a>, <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><b>hence </b><a NAME="E5:101_1_2_1_1_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="newton.html#T11" target="_self">Th11</a>, <a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:101_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div>
