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

<b>let </b><font color="Maroon">c<sub>1</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>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:41"/><i><font color="Green">E34</font></i>: 
(  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is    <a href="supinf_1.html#M2">minorant</a> of  <a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</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">minorant</a> of  <a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</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">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17">SUPINF_1:def 17</a></i>;<br/>
<a NAME="E2:41"/><i><font color="Green">E35</font></i>: 
(  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font> is    <a href="supinf_1.html#M1">majorant</a> of <font color="Maroon">c<sub>1</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#M1">majorant</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/> <a href="supinf_1.html#K6">sup</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> ) )
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/>
<a NAME="E3:41"/><i><font color="Green">E36</font></i>: 
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<a NAME="E1:41_1"/>
 <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is    <a href="supinf_1.html#M1">majorant</a> of  <a href="supinf_2.html#K6" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:41"><i><font color="Green">E34</font></i></a>, <a class="ref" href="supinf_2.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E2:41_1"/><b>then </b>
 <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is    <a href="supinf_1.html#M1">majorant</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E3:41_1"/><b>then </b>
 <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/>
<a NAME="E4:41_1"/><b>then </b>
 <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3" target="_self">-</a> <span class="p2">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p3">(<span class="default"><a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E5:41_1"/>
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:41"/>
 <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<a NAME="E1:41_2"/>
 <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is    <a href="supinf_1.html#M2">minorant</a> of  <a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E2:41"><i><font color="Green">E35</font></i></a>, <a class="ref" href="supinf_2.html#T30" target="_self">Th30</a></i>;<br/>
<b>hence </b><a NAME="E2:41_2"/>
 <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#D17">SUPINF_1:def 17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:41"/>
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K6" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="supinf_2.html#E3:41"><i><font color="Green">E36</font></i></a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>


</div>
