<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>consider </b><font color="Maroon" title="c2">M</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> <b> such that </b><br/><a NAME="E2:8"/><i><font color="Green" title="E9">A1</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">A</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="measure4.html#T8" target="_self" title="MEASURE4:th.8">Th8</a></i>;<br/>
<a NAME="E3:8"/><i><font color="Green" title="E10">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds   <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="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">A</font>
 
<b>by </b><i><a class="txt" href="measure4.html#E2:8"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<a NAME="E4:8"/><b>then </b><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c2">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#D4" title="MEASURE1:def.4">MEASURE1:def 4</a></i>;<br/>
<a NAME="E5:8"/><i><font color="Green" title="E12">A4</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E6:8"/><i><font color="Green" title="E13">A5</font></i>: 
for <font color="Olive" title="b1">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> , <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">F</font> be   <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> , <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font>;<br/>

<a NAME="E1:8_1"/><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">F</font></span>)</span></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="measure4.html#E2:8"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<a NAME="E2:8_1"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">F</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="measure4.html#E4:8"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="measure1.html#T54" title="MEASURE1:th.54">MEASURE1:54</a></i>;<br/>
<a NAME="E3:8_1"/><i><font color="Green" title="E16">A8</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 0 <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="c2">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</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="8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="E1:8_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:8_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c4">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1_1"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c4">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="txt" href="measure4.html#E2:8"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:8_1"/><b>then </b><i><font color="Green" title="E17">A9</font></i>: 
 <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">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#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0
 
<b>by </b><i><a class="txt" href="measure4.html#E2:8_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="supinf_2.html#T67" title="SUPINF_2:th.67">SUPINF_2:67</a></i>;<br/>
<a NAME="E5:8_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K20" title="SUPINF_2:func.20">Ser</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> 0
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63" title="SUPINF_2:th.63">SUPINF_2:63</a></i>;<br/>
<b>hence </b><a NAME="E6:8_1"/>
<font color="Maroon" title="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="measure4.html#E1:8_1"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="measure4.html#E3:8_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="measure4.html#E4:8_1"><i><font color="Green" title="E17">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c2">M</font>
;<br/>

<a NAME="E7:8"/>
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b2">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>

<a NAME="E1:8_2"/>
( <font color="Maroon" title="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c4">B</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="txt" href="measure4.html#E2:8"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:8_2"/>
<font color="Maroon" title="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c3">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Maroon" title="c4">B</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:8"/>
( <font color="Maroon" title="c2">M</font> is <a href="supinf_2.html#V6" title="SUPINF_2:attr.6">nonnegative</a> &amp; <font color="Maroon" title="c2">M</font> <a href="supinf_2.html#K15" title="SUPINF_2:func.15">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; ( for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font> holds <br/>( <font color="Maroon" title="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <font color="Olive" title="b2">B</font> &amp; ( for <font color="Olive" title="b3">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> , <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c2">M</font> <a href="measure1.html#K1" title="MEASURE1:func.1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b3">F</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K21" title="SUPINF_2:func.21">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">F</font></span>)</span> ) ) ) )
 <b>by </b><i><a class="txt" href="measure4.html#E2:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="measure4.html#E3:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="measure4.html#E5:8"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="measure4.html#E6:8"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="measure1.html#D4" title="MEASURE1:def.4">MEASURE1:def 4</a></i>;<br/>


</div>
