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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:64"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">X</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 <b>and </b><br/><a NAME="E2:64"/><i><font color="Green" title="E44">A2</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D3" target="_self" title="PSCOMP_1:def.3">PSCOMP_1:def 3</a>,<a class="ref" href="pscomp_1.html#D14" target="_self" title="PSCOMP_1:def.14">PSCOMP_1:def 14</a><br/>

<a NAME="E3:64"/><i><font color="Green" title="E45">A3</font></i>: 
 <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T40" target="_self" title="PSCOMP_1:th.40">Th40</a></i>;<br/>
<b>hence </b><a NAME="E4:64"/><i><font color="Green" title="E46">A4</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">X</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:64"><i><font color="Green" title="E43">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E22"><i><font color="Green" title="E14">Lm1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D4" target="_self" title="PSCOMP_1:def.4">PSCOMP_1:def 4</a>,<a class="ref" href="pscomp_1.html#D15" target="_self" title="PSCOMP_1:def.15">PSCOMP_1:def 15</a><br/>

<a NAME="E5:64"/><i><font color="Green" title="E47">A5</font></i>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E2:64"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">X</font></span>)</span> = 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pscomp_1.html#E3:64"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="pscomp_1.html#E4:64"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="pscomp_1.html#T17" target="_self" title="PSCOMP_1:th.17">Th17</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E7:64"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E5:64"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="pscomp_1.html#T40" target="_self" title="PSCOMP_1:th.40">Th40</a></i>;<br/>


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