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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="supinf_1.html#M3" target="_self">bool_DOMAIN</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<b>assume </b><a NAME="E1:85"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:85"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K8" target="_self">SUP</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:85_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:85_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:85"><i><font color="Green">E57</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:85_1"/>
<font color="Maroon">c<sub>1</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="numbers.html#K6">ExtREAL</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#D18" target="_self">Def18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="txt" href="supinf_1.html#E3:85_1"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:85_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="txt" href="supinf_1.html#E7:85_1"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E9:85_1"/>
 <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E10:85_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E3:85_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="supinf_1.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E11:85_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K8" target="_self">SUP</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E3:85_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="supinf_1.html#E7:85_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="supinf_1.html#D19" target="_self">Def19</a></i>;<br/>
<b>hence </b><a NAME="E12:85_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6" target="_self">sup</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K8" target="_self">SUP</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="supinf_1.html#E7:85_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="supinf_1.html#E10:85_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="supinf_1.html#T97" target="_self">Th97</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:85"/>
 <a href="supinf_1.html#K6" target="_self">sup</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K8" target="_self">SUP</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is    <a href="supinf_1.html#M1" target="_self">majorant</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D9" target="_self">Def9</a></i>;<br/>


</div>
