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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  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> ;<br/>

<b>assume </b><a NAME="E1:72"/><i><font color="Green">E49</font></i>: 
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/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:72"/>
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/> <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_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="E1:72_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="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:72_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:72"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E4:72_1"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E3:72_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="supinf_1.html#T79" target="_self">Th79</a></i>;<br/>
<b>hence </b><a NAME="E5:72_1"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E3:72_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="supinf_1.html#T29" target="_self">Th29</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:72"/><b>then </b>
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>1</sub></font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E4:72"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D17" target="_self">Def17</a></i>;<br/>


</div>
