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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">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:65"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>and </b><br/><a NAME="E2:65"/><i><font color="Green">E31</font></i>: 
 <a href="pscomp_1.html#K3" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D4" target="_self">PSCOMP_1:def 4</a>,<a class="ref" href="pscomp_1.html#D15" target="_self">PSCOMP_1:def 15</a><br/>

<a NAME="E3:65"/><i><font color="Green">E32</font></i>: 
 <a href="pscomp_1.html#K4" target="_self">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:65"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D3" target="_self">PSCOMP_1:def 3</a>,<a class="ref" href="pscomp_1.html#D14" target="_self">PSCOMP_1:def 14</a><br/>

<a NAME="E5:65"/><i><font color="Green">E35</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4" target="_self">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
 <a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> = 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" target="_self">inf</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K4" target="_self">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K4" target="_self">-</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" target="_self">inf</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E7:65"/>
 <a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


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