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<b>let </b><font color="Maroon">Omega</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">Sigma</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">Sigma</font>;<br/>





<b>set </b><font color="Maroon">Z</font> = <font color="Maroon">Sigma</font>;<br/>
<a NAME="E1:19"/>
 <a href="supinf_1.html#K3">REAL</a>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K6">ExtREAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Sigma</font>, <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E3:19"/>
for <font color="Olive">x</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P1</font> holds <br/> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<a NAME="E1:19_1"/><i><font color="Green">E24</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>assume </b><a NAME="E2:19_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:19_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">Sigma</font> <b>by </b><i>, <a class="ref" href="prob_1.html#D10">PROB_1:def 10</a></i>;<br/>
<a NAME="E6:19_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D13">PROB_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E7:19_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:19"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">P1</font> is <a href="supinf_2.html#V5">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D15">SUPINF_2:def 15</a></i>;<br/>
<a NAME="E5:19"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">P1</font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="ref" href="supinf_2.html#D22">SUPINF_2:def 22</a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="ref" href="prob_1.html#T64">PROB_1:64</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/>
<a NAME="E7:19"/>
for <font color="Olive">F</font> being  <a href="measure1.html#NM4">Sep_Sequence</a> of <font color="Maroon">Sigma</font> holds   <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="partfun1.html#K1">*</a> <font color="Olive">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="measure1.html#K5">rng</a> <font color="Olive">F</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be   <a href="measure1.html#NM4">Sep_Sequence</a> of <font color="Maroon">Sigma</font>;<br/>

<b>reconsider </b><font color="Maroon">F2</font> = <font color="Maroon">F</font> as  <a href="prob_2.html#V1">V129</a>  <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">Sigma</font> <b>by </b><i/>;<br/>
<a NAME="E2:19_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">F2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">F2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_3.html#T51">PROB_3:51</a></i>;<br/>
<a NAME="E3:19_2"/>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">F2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:19_2_1"/>
 <a href="series_1.html#K1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">F2</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="prob_3.html#T50">PROB_3:50</a></i>;<br/>
<a NAME="E2:19_2_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">F2</font> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="series_1.html#D2">SERIES_1:def 2</a></i>;<br/>
<a NAME="E3:19_2_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="prob_3.html#T4">PROB_3:4</a></i>;<br/>
<a NAME="E4:19_2_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">F2</font> is <a href="partfun3.html#V4">nonnegative</a>
 
<b>by </b><i><a class="ref" href="rinfsup1.html#D3">RINFSUP1:def 3</a></i>;<br/>
<b>thus </b><a NAME="E5:19_2_1"/>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">F2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:19_2"/>
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="measure1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:19"/>
<font color="Maroon">P</font> is    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">Sigma</font>
 <b>by </b><i>, , <a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/>


</div>
