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

<b>set </b><font color="Maroon">x</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span> ;<br/>
<a NAME="E1:1"/><i><font color="Green">E36</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</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="E3:1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 3 <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>
<a NAME="E4:1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 3
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:1_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 )
 
<b>by </b><i><a class="ref" href="cqc_the1.html#T4">CQC_THE1:4</a></i>;<br/>
<b>hence </b><a NAME="E6:1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:1"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>
 ;<br/>

<a NAME="E2:1_2"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 3 )
 
<b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>thus </b>4 = 
<span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 4 </span>}</span> 
<b>by </b><i><a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
;<br/>


</div>
