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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be    <a href="measure1.html#M3" title="MEASURE1:mode.3">sigma_Measure</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">T</font> be    <a href="measure2.html#M1" title="MEASURE2:mode.1">N_Measure_fam</a> of <font color="Maroon" title="c2">S</font>;<br/>







<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E14">A1</font></i>: 
for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font> holds <br/><font color="Olive" title="b1">A</font> is    <a href="measure1.html#M4" title="MEASURE1:mode.4">measure_zero</a> of <font color="Maroon" title="c3">M</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:16"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="ref" href="measure2.html#T14" target="_self" title="MEASURE2:th.14">Th14</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">G</font> = <font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font>;<br/>
<a NAME="E4:16"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure2.html#T1" target="_self" title="MEASURE2:th.1">Th1</a></i>;<br/>
<a NAME="E5:16"/><i><font color="Green" title="E17">A4</font></i>: 
for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <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="Olive" title="b1">r</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">r</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:16_1"/>
<font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">r</font> is    <a href="measure1.html#M4" title="MEASURE1:mode.4">measure_zero</a> of <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="measure2.html#E1:16"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="measure2.html#E3:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>
<a NAME="E2:16_1"/><b>then </b>
<font color="Maroon" title="c3">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="ref" href="measure1.html#D13" title="MEASURE1:def.13">MEASURE1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E3:16_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="c7">r</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:16"/><b>then </b>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K19" title="SUPINF_2:func.19">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="measure2.html#E4:16"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="supinf_2.html#T67" title="SUPINF_2:th.67">SUPINF_2:67</a></i>;<br/>
<a NAME="E7:16"/><b>then </b>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63" title="SUPINF_2:th.63">SUPINF_2:63</a></i>;<br/>
<a NAME="E8:16"/><b>then </b>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="txt" href="measure2.html#E5:16"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E9:16"/><b>then </b><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="txt" href="measure2.html#E3:16"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="measure2.html#T13" target="_self" title="MEASURE2:th.13">Th13</a></i>;<br/>
<a NAME="E10:16"/>
<font color="Maroon" title="c3">M</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="measure1.html#D11" title="MEASURE1:def.11">MEASURE1:def 11</a></i>;<br/>
<a NAME="E11:16"/><b>then </b>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c4">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="measure1.html#D4" title="MEASURE1:def.4">MEASURE1:def 4</a></i>;<br/>
<a NAME="E12:16"/><b>then </b>
<font color="Maroon" title="c3">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 
<b>by </b><i><a class="txt" href="measure2.html#E9:16"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E13:16"/>
 <a href="measure2.html#K2" title="MEASURE2:func.2">union</a> <font color="Maroon" title="c4">T</font> is    <a href="measure1.html#M4" title="MEASURE1:mode.4">measure_zero</a> of <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#D13" title="MEASURE1:def.13">MEASURE1:def 13</a></i>;<br/>


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