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

<b>let </b><font color="Maroon">IT</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> , <font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>



<b>given </b><font color="Maroon">X0</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> <b> such that </b><a NAME="E1:12_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">X0</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Olive">r</font> <a href="real_1.html#K5">-</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E2:12_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X0</font>
 ;<br/>

<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Olive">r</font> <a href="real_1.html#K5">-</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span> ;<br/>
<a NAME="E3:12_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Olive">r</font> <a href="real_1.html#K5">-</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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="12_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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="E1:12_1_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Olive">r</font> <a href="real_1.html#K5">-</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:12_1_2_1"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="real_1.html#K5">-</a> <font color="Olive">s</font></span>)</span> &amp; <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
;<br/>
<b>hence </b><a NAME="E3:12_1_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y0</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Olive">r</font> <a href="real_1.html#K5">-</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>take </b>
<font color="Maroon">Y0</font>
;<br/>

<b>thus </b><a NAME="E5:12_1_2"/>
<font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Olive">r</font> <a href="real_1.html#K5">-</a> <font color="Olive">s</font></span>)</span></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span> 
 ;<br/>

<a NAME="E6:12_1_2"/>
<font color="Maroon">X0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_1_2_2"/>
<font color="Maroon">X0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y0</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:12_1_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X0</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:12_1_2_2_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:12_1_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/>
<b>hence </b><a NAME="E5:12_1_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">x</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="E2:12_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y0</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:12_1_2_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>
<a NAME="E5:12_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/>
<b>hence </b><a NAME="E6:12_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X0</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:12_1_2"/>
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Y0</font>
 <b>by </b><i/>;<br/>


</div>
