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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K4" title="SUPINF_1:func.4">SetMajorant</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp;  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K5" title="SUPINF_1:func.5">SetMinorant</a> <font color="Maroon" title="c1">X</font></span>)</span> )</span><br/>

<a NAME="E1:52"/>
 <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c1">X</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D16" target="_self" title="SUPINF_1:def.16">Def16</a></i>;<br/>
<a NAME="E2:52"/><b>then </b><i><font color="Green" title="E38">A1</font></i>: 
 <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K4" title="SUPINF_1:func.4">SetMajorant</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D14" target="_self" title="SUPINF_1:def.14">Def14</a></i>;<br/>
<a NAME="E3:52"/>
for <font color="Olive" title="b1">y</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K4" title="SUPINF_1:func.4">SetMajorant</a> <font color="Maroon" title="c1">X</font> holds <br/> <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c1">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">y</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K4" title="SUPINF_1:func.4">SetMajorant</a> <font color="Maroon" title="c1">X</font> implies  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <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> )</span><br/>

<b>assume </b><a NAME="E1:52_1"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K4" title="SUPINF_1:func.4">SetMajorant</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <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></span><br/>

<a NAME="E2:52_1"/><b>then </b>
<font color="Maroon" title="c2">y</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D14" target="_self" title="SUPINF_1:def.14">Def14</a></i>;<br/>
<b>hence </b><a NAME="E3:52_1"/>
 <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <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>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16" target="_self" title="SUPINF_1:def.16">Def16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:52"/><b>then </b><i><font color="Green" title="E39">A2</font></i>: 
 <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c1">X</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of  <a href="supinf_1.html#K4" title="SUPINF_1:func.4">SetMajorant</a> <font color="Maroon" title="c1">X</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="E5:52"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c1">X</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c1">X</font>
 
<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="E6:52"/><b>then </b><i><font color="Green" title="E40">A3</font></i>: 
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K5" title="SUPINF_1:func.5">SetMinorant</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D15" target="_self" title="SUPINF_1:def.15">Def15</a></i>;<br/>
<a NAME="E7:52"/>
for <font color="Olive" title="b1">y</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K5" title="SUPINF_1:func.5">SetMinorant</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">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> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">y</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K5" title="SUPINF_1:func.5">SetMinorant</a> <font color="Maroon" title="c1">X</font> implies <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> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K5" title="SUPINF_1:func.5">SetMinorant</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:52_2"/><b>then </b>
<font color="Maroon" title="c2">y</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D15" target="_self" title="SUPINF_1:def.15">Def15</a></i>;<br/>
<b>hence </b><a NAME="E3:52_2"/>
<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> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D17" target="_self" title="SUPINF_1:def.17">Def17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:52"/><b>then </b>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c1">X</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of  <a href="supinf_1.html#K5" title="SUPINF_1:func.5">SetMinorant</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9" target="_self" title="SUPINF_1:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E9:52"/>
(  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K4" title="SUPINF_1:func.4">SetMajorant</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp;  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K5" title="SUPINF_1:func.5">SetMinorant</a> <font color="Maroon" title="c1">X</font></span>)</span> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E2:52"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="supinf_1.html#E4:52"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="supinf_1.html#E6:52"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="supinf_1.html#T80" target="_self" title="SUPINF_1:th.80">Th80</a>, <a class="ref" href="supinf_1.html#T81" target="_self" title="SUPINF_1:th.81">Th81</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
