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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">LB1</font> be    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">LB2</font> be    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c2">Y</font>;<br/>





<a NAME="E1:70"/>
for <font color="Olive" title="b1">a</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font> holds <br/> <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:70_1"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font>
 ;<br/>

<a NAME="E2:70_1"/><i><font color="Green" title="E47">A2</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:70_1_1"/><b>then </b><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c3">LB1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D10" target="_self" title="SUPINF_1:def.10">Def10</a></i>;<br/>
<a NAME="E3:70_1_1"/><i><font color="Green" title="E49">A4</font></i>: 
<font color="Maroon" title="c3">LB1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E4:70_1_1"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17" target="_self" title="SUPINF_1:def.17">Def17</a></i>;<br/>
<a NAME="E5:70_1_1"/><b>then </b>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">LB1</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E3:70_1_1"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="supinf_1.html#D10" target="_self" title="SUPINF_1:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E6:70_1_1"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E2:70_1_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:70_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> implies  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 ;<br/>

<a NAME="E2:70_1_2"/><b>then </b><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon" title="c4">LB2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D10" target="_self" title="SUPINF_1:def.10">Def10</a></i>;<br/>
<a NAME="E3:70_1_2"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c4">LB2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E4:70_1_2"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D17" target="_self" title="SUPINF_1:def.17">Def17</a></i>;<br/>
<a NAME="E5:70_1_2"/><b>then </b>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">LB2</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E3:70_1_2"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="supinf_1.html#D10" target="_self" title="SUPINF_1:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E6:70_1_2"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E2:70_1_2"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:70_1"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:70_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="supinf_1.html#E2:70_1"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:70"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">LB1</font>,<font color="Maroon" title="c4">LB2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D10" target="_self" title="SUPINF_1:def.10">Def10</a></i>;<br/>


</div>
