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<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="supinf_1.html#NM6" title="SUPINF_1:NM.6">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 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="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Olive" title="b1">S</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of  <a href="supinf_1.html#K8" title="SUPINF_1:func.8">SUP</a> <font color="Maroon" title="c1">F</font></span><br/><b>let </b><font color="Maroon" title="c2">S</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"> ( <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="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">S</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of  <a href="supinf_1.html#K8" title="SUPINF_1:func.8">SUP</a> <font color="Maroon" title="c1">F</font> )</span><br/>



<b>assume </b><a NAME="E1:79"/><i><font color="Green" title="E50">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="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">S</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of  <a href="supinf_1.html#K8" title="SUPINF_1:func.8">SUP</a> <font color="Maroon" title="c1">F</font></span><br/>

<a NAME="E2:79"/>
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>  <a href="supinf_1.html#K8" title="SUPINF_1:func.8">SUP</a> <font color="Maroon" title="c1">F</font> holds <br/><font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_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>  <a href="supinf_1.html#K8" title="SUPINF_1:func.8">SUP</a> <font color="Maroon" title="c1">F</font> implies <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">S</font> )</span><br/>

<b>assume </b><a NAME="E1:79_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K8" title="SUPINF_1:func.8">SUP</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"> <font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">S</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">A</font> being  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> such that </b><br/><a NAME="E3:79_1"/><i><font color="Green" title="E51">A2</font></i>: 
( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> &amp; <font color="Maroon" title="c3">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> <font color="Maroon" title="c4">A</font> )
 <b>by </b><i><a class="ref" href="supinf_1.html#D19" target="_self" title="SUPINF_1:def.19">Def19</a></i>;<br/>
<a NAME="E4:79_1"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="membered.html#D8" title="MEMBERED:def.8">MEMBERED:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> )</span><br/>

<b>assume </b><a NAME="E1:79_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font></span><br/>

<b>hence </b><a NAME="E2:79_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:79"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="supinf_1.html#E3:79_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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="E5:79_1"/>
<font color="Maroon" title="c3">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E3:79_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="supinf_1.html#T91" target="_self" title="SUPINF_1:th.91">Th91</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="E3:79"/>
 <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">S</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of  <a href="supinf_1.html#K8" title="SUPINF_1:func.8">SUP</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D9" target="_self" title="SUPINF_1:def.9">Def9</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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