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<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="supinf_1.html#NM4" title="SUPINF_1:NM.4">bool_DOMAIN</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">F</font> holds <br/> <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6" title="SUPINF_1:func.6">INF</a> <font color="Maroon" title="c1">F</font></span>)</span> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Olive" title="b1">S</font></span><br/><b>let </b><font color="Maroon" title="c2">S</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">F</font> implies  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6" title="SUPINF_1:func.6">INF</a> <font color="Maroon" title="c1">F</font></span>)</span> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c2">S</font> )</span><br/>



<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c1">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6" title="SUPINF_1:func.6">INF</a> <font color="Maroon" title="c1">F</font></span>)</span> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E2:30"/>
for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> holds <br/> <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6" title="SUPINF_1:func.6">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="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> implies  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6" title="SUPINF_1:func.6">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> )</span><br/>

<b>assume </b><a NAME="E1:30_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6" title="SUPINF_1:func.6">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></span><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="E3:30_1"/><i><font color="Green" title="E12">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="#E1:30"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>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="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="#E3:30_1"><i><font color="Green" title="E12">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="E6:30_1"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">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="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="#E6:30_1"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/>
<a NAME="E8:30_1"/><i><font color="Green" title="E14">A4</font></i>: 
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <font color="Maroon" title="c5">Z</font> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c5">Z</font>
 
<b>by </b><i><a class="ref" href="xxreal_2.html#D4" title="XXREAL_2:def.4">XXREAL_2:def 4</a></i>;<br/>
<a NAME="E9:30_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">INF</a> <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="txt" href="#E3:30_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E6:30_1"><i><font color="Green" title="E13">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="E10:30_1"/>
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6" title="SUPINF_1:func.6">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="#E6:30_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="#E8:30_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="#E3:30_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="xxreal_2.html#T62" title="XXREAL_2:th.62">XXREAL_2:62</a>, <a class="ref" href="xxreal_2.html#D2" title="XXREAL_2:def.2">XXREAL_2:def 2</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>
<b>hence </b><a NAME="E3:30"/>
 <a href="xxreal_2.html#K2" title="XXREAL_2:func.2">inf</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6" title="SUPINF_1:func.6">INF</a> <font color="Maroon" title="c1">F</font></span>)</span> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="xxreal_2.html#D2" title="XXREAL_2:def.2">XXREAL_2:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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