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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon">F</font> is <a href="supinf_2.html#V2" target="_self">bounded_below</a>
 ;<br/><a NAME="E2:58_1"/><b>then </b>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E3:58_1"/><b>then </b>
 <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">F</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K3" target="_self">-</a> <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i/>;<br/><a NAME="E4:58_1"/><b>then </b>
 <a href="supinf_2.html#K8" target="_self">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K12" target="_self">-</a> <font color="Maroon">F</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E5:58_1"/>
 <a href="supinf_2.html#K12" target="_self">-</a> <font color="Maroon">F</font> is <a href="supinf_2.html#V1" target="_self">bounded_above</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T12" target="_self">Th12</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:58"/>
 <a href="supinf_2.html#K12" target="_self">-</a> <font color="Maroon">F</font> is <a href="supinf_2.html#V1" target="_self">bounded_above</a>
 ;<br/>

<a NAME="E3:58"/><b>then </b>
 <a href="supinf_2.html#K8" target="_self">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K12" target="_self">-</a> <font color="Maroon">F</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_2.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E4:58"/><b>then </b>
 <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">F</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_2.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E5:58"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="supinf_2.html#K3" target="_self">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3" target="_self">-</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:58"/>
(  <a href="supinf_2.html#K3" target="_self">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_2.html#K3" target="_self">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3" target="_self">-</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">F</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:58"/>
<font color="Maroon">F</font> is <a href="supinf_2.html#V2" target="_self">bounded_below</a>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T13" target="_self">Th13</a></i>;<br/>


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