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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="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">c</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:32_1"/><i><font color="Green" title="E31">A2</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">c</font>
 ;<br/><b>set </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c2">f</font>;<br/><a NAME="E2:32_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Olive" title="b1">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_1_1"/><i><font color="Green" title="E32">A3</font></i>: 
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/>

<a NAME="E2:32_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:32"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/>
<a NAME="E3:32_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E1:32_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="extreal2.html#T45" title="EXTREAL2:th.45">EXTREAL2:45</a></i>;<br/>
<b>hence </b><a NAME="E4:32_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:32_1_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:32_1"/>
<font color="Maroon" title="c3">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c2">f</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T71" title="SUPINF_2:th.71">SUPINF_2:71</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:32"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/>

<b>set </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c2">f</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:32_2"/><i><font color="Green" title="E32">A5</font></i>: 
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/><a NAME="E2:32_2"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:32"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="supinf_2.html#T70" title="SUPINF_2:th.70">SUPINF_2:70</a></i>;<br/><a NAME="E3:32_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E3:32"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="extreal2.html#T46" title="EXTREAL2:th.46">EXTREAL2:46</a></i>;<br/><b>hence </b><a NAME="E4:32_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mesfunc1.html#K4" title="MESFUNC1:func.4">.</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:32_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="mesfunc1.html#D6" title="MESFUNC1:def.6">MESFUNC1:def 6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:32"/>
<font color="Maroon" title="c3">c</font> <a href="mesfunc1.html#K8" title="MESFUNC1:func.8">(#)</a> <font color="Maroon" title="c2">f</font> is <a href="mesfunc5.html#V2" title="MESFUNC5:attr.2">nonpositive</a>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T15" target="_self" title="MESFUNC5:th.15">Th15</a></i>;<br/>


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