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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="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="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Olive" title="b1">S</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#K8" title="SUPINF_1:func.8">SUP</a> <font color="Maroon" title="c1">F</font></span>)</span></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> <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#K8" title="SUPINF_1:func.8">SUP</a> <font color="Maroon" title="c1">F</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E51">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> <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#K8" title="SUPINF_1:func.8">SUP</a> <font color="Maroon" title="c1">F</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c3">a</font> =  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">S</font>;<br/>
<b>set </b><font color="Maroon" title="c4">b</font> =  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K8" title="SUPINF_1:func.8">SUP</a> <font color="Maroon" title="c1">F</font></span>)</span>;<br/>
<a NAME="E2:81"/><i><font color="Green" title="E52">A2</font></i>: 
 <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K8" title="SUPINF_1:func.8">SUP</a> <font color="Maroon" title="c1">F</font></span>)</span> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:81"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="supinf_1.html#T113" target="_self" title="SUPINF_1:th.113">Th113</a></i>;<br/>
<a NAME="E3:81"/>
 <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="txt" href="supinf_1.html#E1:81"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="supinf_1.html#T112" target="_self" title="SUPINF_1:th.112">Th112</a></i>;<br/>
<a NAME="E4:81"/><b>then </b>
(  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">S</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> <span class="p1">(<span class="default"><a href="supinf_1.html#K8" title="SUPINF_1:func.8">SUP</a> <font color="Maroon" title="c1">F</font></span>)</span> &amp;  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K8" title="SUPINF_1:func.8">SUP</a> <font color="Maroon" title="c1">F</font></span>)</span> <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#E2:81"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="supinf_1.html#D16" target="_self" title="SUPINF_1:def.16">Def16</a></i>;<br/>
<b>hence </b><a NAME="E5:81"/>
 <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c2">S</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#K8" title="SUPINF_1:func.8">SUP</a> <font color="Maroon" title="c1">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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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