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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="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E37</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font>
 ;<br/><b>set </b><font color="Maroon">g</font> = <font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font>;<br/><a NAME="E2:32_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<a NAME="E2:32_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<br/>
<a NAME="E3:32_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="extreal2.html#T45">EXTREAL2:45</a></i>;<br/>
<b>hence </b><a NAME="E4:32_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T2" target="_self">Th2</a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/>


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

<b>assume </b><a NAME="E3:32"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<b>set </b><font color="Maroon">g</font> = <font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E2:32_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T70">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">R_EAL</a> <font color="Maroon">c</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i>, <a class="ref" href="extreal2.html#T46">EXTREAL2:46</a></i>;<br/><b>hence </b><a NAME="E4:32_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="mesfunc5.html#T2" target="_self">Th2</a>, <a class="ref" href="mesfunc1.html#D6">MESFUNC1:def 6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:32"/>
<font color="Maroon">c</font> <a href="mesfunc1.html#K8">(#)</a> <font color="Maroon">f</font> is <a href="mesfunc5.html#V2" target="_self">nonpositive</a>
 <b>by </b><i/>;<br/>


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