<?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="E1:66"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:66"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>2</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>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:66_1"/>
(  <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">c<sub>2</sub></font> &amp; ( 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> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <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#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/>
<a NAME="E2:66_1"/><b>then </b>
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:66"><i><font color="Green">E46</font></i></a>, <a class="ref" href="supinf_1.html#T39" target="_self">Th39</a></i>;<br/>
<b>hence </b><a NAME="E3:66_1"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>2</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>1</sub></font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D17" target="_self">Def17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:66"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <font color="Maroon">c<sub>2</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>1</sub></font>
 ;<br/>


</div>
