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

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span> where B is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> ;<br/>
<a NAME="E1:62_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span> where B is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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="E1:62_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span> where B is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:62_1_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:62_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:62_1_1_1"><i><font color="Green">E59</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:62_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span> where B is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div>
