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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> ; <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">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>   st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">f</font> is <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> holds <br/><font color="Olive" title="b1">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a></span><br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <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"> (  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> is <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> implies <font color="Maroon" title="c2">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> )</span><br/>



<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E20">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> is <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</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">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a></span><br/>

<a NAME="E2:32"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>
 
<b>by </b><i><a class="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">UB</font> being    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font><b> such that </b><br/><a NAME="E4:32"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c3">UB</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#D11" title="SUPINF_1:def.11">SUPINF_1:def 11</a></i>;<br/>
<a NAME="E5:32"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> is <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>
 
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="xxreal_2.html#D11" title="XXREAL_2:def.11">XXREAL_2:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">LB</font> being    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font><b> such that </b><br/><a NAME="E7:32"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c4">LB</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#D12" title="SUPINF_1:def.12">SUPINF_1:def 12</a></i>;<br/>
<a NAME="E8:32"/><i><font color="Green" title="E23">A4</font></i>: 
( <font color="Maroon" title="c3">UB</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp;  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">LB</font> )
 
<b>by </b><i><a class="txt" href="#E4:32"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="#E7:32"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> implies <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )</span><br/><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> </span><br/><a NAME="E2:32_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:32_1"/><b>then </b>
( <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">UB</font> &amp; <font color="Maroon" title="c4">LB</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="ref" href="xxreal_2.html#D1" title="XXREAL_2:def.1">XXREAL_2:def 1</a>, <a class="ref" href="xxreal_2.html#D2" title="XXREAL_2:def.2">XXREAL_2:def 2</a></i>;<br/><a NAME="E4:32_1"/><b>then </b><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp;  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">x</font> )
 <b>by </b><i><a class="txt" href="#E8:32"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:32_1"/><b>then </b>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T4" title="EXTREAL1:th.4">EXTREAL1:4</a></i>;<br/><b>hence </b><a NAME="E6:32_1"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="#E4:32_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="extreal2.html#T59" title="EXTREAL2:th.59">EXTREAL2:59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:32"/>
<font color="Maroon" title="c2">f</font> is <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>
 <b>by </b><i><a class="ref" href="mesfunc2.html#D1" title="MESFUNC2:def.1">MESFUNC2:def 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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