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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="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E24">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="E1:63_1"/><i><font color="Green" title="E25">A2</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:63_1_1"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c4">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">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/><a NAME="E2:63_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="mesfunc6.html#E1:63"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="mesfunc6.html#T51" target="_self" title="MESFUNC6:th.51">Th51</a></i>;<br/><a NAME="E3:63_1_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc6.html#E1:63_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E4:63_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="mesfunc6.html#E1:63_1_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:63_1"/>
<font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</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="mesfunc6.html#T52" target="_self" title="MESFUNC6:th.52">Th52</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:63"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:63_2"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c4">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">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/><a NAME="E2:63_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="mesfunc6.html#E1:63"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="mesfunc6.html#T51" target="_self" title="MESFUNC6:th.51">Th51</a></i>;<br/><a NAME="E3:63_2"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="mesfunc6.html#E3:63"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="xreal_1.html#T67" title="XREAL_1:th.67">XREAL_1:67</a></i>;<br/><b>hence </b><a NAME="E4:63_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="mesfunc6.html#E1:63_2"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:63"/>
<font color="Maroon" title="c3">r</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</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="mesfunc6.html#T54" target="_self" title="MESFUNC6:th.54">Th54</a></i>;<br/>


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