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<div class="add">

<b>let </b><font color="Maroon" title="c1">F</font> be    <a href="supinf_1.html#M3" title="SUPINF_1:mode.3">bool_DOMAIN</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/>



<b>assume </b><a NAME="E1:90"/><i><font color="Green" title="E60">A1</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">F</font>
 ;<br/>

<a NAME="E2:90"/>
for <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> holds <br/> <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K9" title="SUPINF_1:func.9">INF</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:90_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:90_1"/><i><font color="Green" title="E61">A2</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> &amp; <font color="Maroon" title="c4">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:90"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:90_1"/>
<font color="Maroon" title="c1">F</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#D18" target="_self" title="SUPINF_1:def.18">Def18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">Z</font> = <font color="Maroon" title="c4">Z</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  <b>by </b><i><a class="txt" href="supinf_1.html#E3:90_1"><i><font color="Green" title="E61">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:90_1"/><i><font color="Green" title="E62">A3</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c5">Z</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">a</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="txt" href="supinf_1.html#E7:90_1"><i><font color="Green" title="E62">A3</font></i></a></i>;<br/>
<a NAME="E9:90_1"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c5">Z</font> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c5">Z</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="E10:90_1"/><b>then </b><i><font color="Green" title="E63">A4</font></i>: 
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <font color="Maroon" title="c5">Z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E3:90_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="supinf_1.html#D10" target="_self" title="SUPINF_1:def.10">Def10</a></i>;<br/>
<a NAME="E11:90_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K9" title="SUPINF_1:func.9">INF</a> <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E3:90_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="supinf_1.html#E7:90_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="supinf_1.html#D20" target="_self" title="SUPINF_1:def.20">Def20</a></i>;<br/>
<b>hence </b><a NAME="E12:90_1"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K9" title="SUPINF_1:func.9">INF</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E7:90_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="supinf_1.html#E10:90_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="supinf_1.html#T98" target="_self" title="SUPINF_1:th.98">Th98</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:90"/>
 <a href="supinf_1.html#K7" title="SUPINF_1:func.7">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K9" title="SUPINF_1:func.9">INF</a> <font color="Maroon" title="c1">F</font></span>)</span> is    <a href="supinf_1.html#M2" title="SUPINF_1:mode.2">minorant</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D10" target="_self" title="SUPINF_1:def.10">Def10</a></i>;<br/>


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