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

<b>let </b><font color="Maroon" title="c1">Omega</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Sigma</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font><br/> for <font color="Olive" title="b2">P</font> being   <a href="prob_1.html#M3" title="PROB_1:mode.3">Probability</a> of <font color="Olive" title="b1">Sigma</font> holds  <font color="Olive" title="b2">P</font> is   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Olive" title="b1">Sigma</font></span><br/><b>let </b><font color="Maroon" title="c2">Sigma</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <a href="prob_1.html#M3" title="PROB_1:mode.3">Probability</a> of <font color="Maroon" title="c2">Sigma</font> holds  <font color="Olive" title="b1">P</font> is   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">Sigma</font></span><br/><b>let </b><font color="Maroon" title="c3">P</font> be    <a href="prob_1.html#M3" title="PROB_1:mode.3">Probability</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">P</font> is   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">Sigma</font></span><br/>





<b>set </b><font color="Maroon" title="c4">Z</font> = <font color="Maroon" title="c2">Sigma</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">P1</font> = <font color="Maroon" title="c3">P</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">Sigma</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a>, <a class="ref" href="numbers.html#T31" title="NUMBERS:th.31">NUMBERS:31</a></i>;<br/>
<a NAME="E2:18"/>
for <font color="Olive" title="b1">x</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</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#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">P1</font> holds <br/> <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="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">P1</font> implies  <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="c6">x</font> )</span><br/>

<a NAME="E1:18_1"/><i><font color="Green" title="E15">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Sigma</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>assume </b><a NAME="E2:18_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">P1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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="c6">x</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:18_1"/><i><font color="Green" title="E16">A2</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> &amp; <font color="Maroon" title="c5">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="#E1:18_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c7">y</font> as    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font> <b>by </b><i><a class="txt" href="#E4:18_1"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<a NAME="E6:18_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="c3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y1</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D13" title="PROB_1:def.13">PROB_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E7:18_1"/>
 <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="c6">x</font>
 <b>by </b><i><a class="txt" href="#E4:18_1"><i><font color="Green" title="E16">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:18"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">P1</font> is <a href="supinf_2.html#V5" title="SUPINF_2:attr.5">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D15" title="SUPINF_2:def.15">SUPINF_2:def 15</a></i>;<br/>
<a NAME="E4:18"/><b>then </b><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c5">P1</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#D22" title="SUPINF_2:def.22">SUPINF_2:def 22</a></i>;<br/>
<a NAME="E5:18"/>
for <font color="Olive" title="b1">F</font> being  <a href="measure1.html#NM4" title="MEASURE1:NM.4">Sep_Sequence</a> of <font color="Maroon" title="c2">Sigma</font> holds   <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P1</font> <a href="funct_1.html#K1" title="FUNCT_1: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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Olive" title="b1">F</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">F</font> be   <a href="measure1.html#NM4" title="MEASURE1:NM.4">Sep_Sequence</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P1</font> <a href="funct_1.html#K1" title="FUNCT_1: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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">F2</font> = <font color="Maroon" title="c6">F</font> as  <a href="prob_2.html#V1" title="PROB_2:attr.1">V158</a>  <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font> <b>by </b><i><a class="ref" href="prob_4.html#T2" target="_self" title="PROB_4:th.2">Th2</a></i>;<br/>
<a NAME="E2:18_2"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c7">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">F2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_3.html#T51" title="PROB_3:th.51">PROB_3:51</a></i>;<br/>
<a NAME="E3:18_2"/>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_2_1"/>
 <a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">F2</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="prob_3.html#T50" title="PROB_3:th.50">PROB_3:50</a></i>;<br/>
<a NAME="E2:18_2_1"/><b>then </b><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">F2</font> is <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D2" title="SERIES_1:def.2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E3:18_2_1"/>
for <font color="Olive" title="b1">n</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>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">F2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="prob_3.html#T4" title="PROB_3:th.4">PROB_3:4</a></i>;<br/>
<a NAME="E4:18_2_1"/><b>then </b>
<font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">F2</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#D3" title="RINFSUP1:def.3">RINFSUP1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:18_2_1"/>
 <a href="series_1.html#K2" title="SERIES_1:func.2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:18_2_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="prob_4.html#T13" target="_self" title="PROB_4:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2"/>
 <a href="supinf_2.html#K20" title="SUPINF_2:func.20">SUM</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P1</font> <a href="funct_1.html#K1" title="FUNCT_1: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="measure1.html#K4" title="MEASURE1:func.4">rng</a> <font color="Maroon" title="c6">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:18_2"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:18"/>
<font color="Maroon" title="c3">P</font> is   <a href="measure1.html#NM5" title="MEASURE1:NM.5">sigma_Measure</a> of <font color="Maroon" title="c2">Sigma</font>
 <b>by </b><i><a class="txt" href="#E4:18"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="measure1.html#D11" title="MEASURE1:def.11">MEASURE1:def 11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
