<?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="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>thus </b><a NAME="E1:68"/>
( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font>
 ;<br/>

<a NAME="E2:68_1"/><b>then </b><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T95" target="_self" title="SUPINF_1:th.95">Th95</a></i>;<br/>
<a NAME="E3:68_1"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</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="E4:68_1"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</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="E5:68_1"/><b>then </b>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E3:68_1"><i><font color="Green" title="E48">A2</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="E6:68_1"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E2:68_1"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="supinf_1.html#E26"><i><font color="Green" title="E6">Lm3</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:68"/>
<font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 ;<br/>

<a NAME="E3:68"/><b>then </b><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T95" target="_self" title="SUPINF_1:th.95">Th95</a></i>;<br/>
<a NAME="E4:68"/><i><font color="Green" title="E48">A4</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</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="E5:68"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</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="E6:68"/><b>then </b>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E4:68"><i><font color="Green" title="E48">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="E7:68"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E3:68"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="supinf_1.html#E26"><i><font color="Green" title="E6">Lm3</font></i></a></i>;<br/>


</div>
