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

<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:88"/><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">q</font>
 ;<br/>

<a NAME="E2:88"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c1">p</font> holds <br/><font color="Olive" title="b1">x</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="c2">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:88_1"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c3">x</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="c1">p</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">x'</font> = <font color="Maroon" title="c3">x</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="E3:88_1"/>
( <font color="Maroon" title="c4">x'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">x'</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#E1:88_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="newton.html#D7" target="_self" title="NEWTON:def.7">Def7</a></i>;<br/>
<a NAME="E4:88_1"/><b>then </b>
( <font color="Maroon" title="c4">x'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">q</font> &amp; <font color="Maroon" title="c4">x'</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#E1:88"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:88_1"/>
<font color="Maroon" title="c3">x</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="c2">q</font>
 <b>by </b><i><a class="ref" href="newton.html#D7" target="_self" title="NEWTON:def.7">Def7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:88"/>
 <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <font color="Maroon" title="c1">p</font> <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="c2">q</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div>
