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

<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">LB1</font> be    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">LB2</font> be    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">Y</font>;<br/>





<a NAME="E1:74"/>
for <font color="Olive">a</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font> holds <br/> <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:74_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:74_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> implies  <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:74_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">LB1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:74_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">LB1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E4:74_1_1"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E5:74_1_1"/><b>then </b>
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB1</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T51" target="_self">Th51</a>, </i>;<br/>
<b>hence </b><a NAME="E6:74_1_1"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D13" target="_self">Def13</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:74_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> implies  <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:74_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">LB2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:74_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">LB2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E4:74_1_2"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E5:74_1_2"/><b>then </b>
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LB2</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T51" target="_self">Th51</a>, </i>;<br/>
<b>hence </b><a NAME="E6:74_1_2"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D13" target="_self">Def13</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:74_1"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D12" target="_self">Def12</a>, <a class="ref" href="supinf_1.html#T47" target="_self">Th47</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:74"/>
 <a href="supinf_1.html#K7" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">LB1</font>,<font color="Maroon">LB2</font></span><a href="domain_1.html#K7">}</a></span> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div>
