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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/>





<b>assume </b><a NAME="E1:62"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:62_1"/>
<font color="Maroon">F</font> is <a href="supinf_2.html#V1" target="_self">bounded_above</a>
 ;<br/><a NAME="E2:62_1"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">F</font> <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/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><a NAME="E4:62_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp;  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><a NAME="E5:62_1"/><b>then </b>
<font color="Maroon">F</font> <a href="supinf_2.html#K10" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:62_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">F</font> <a href="supinf_2.html#K10" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><a NAME="E7:62_1"/>
<font color="Maroon">F</font> <a href="supinf_2.html#K10" target="_self">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><a NAME="E8:62_1"/><b>then </b><i><font color="Green">E31</font></i>: 
not  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i>, , <a class="ref" href="supinf_1.html#T17">SUPINF_1:17</a></i>;<br/><a NAME="E9:62_1"/>
(  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E10:62_1"/>
 <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:62"/>
 <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>

<a NAME="E4:62"/><b>then </b>
 <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">F</font> <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_1.html#T31">SUPINF_1:31</a></i>;<br/>
<b>hence </b><a NAME="E5:62"/>
<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/>


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