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

<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">ASeq</font> be    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">Sigma</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/>







<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">ASeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:13_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"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0;<br/>
<a NAME="E1:13_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0 = 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> 0
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">ASeq</font> <a href="prob_4.html#K1" target="_self">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="prob_2.html#T11">PROB_2:11</a></i>
<br/>.= 
0
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:13_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="series_1.html#D1">SERIES_1:def 1</a></i>
<br/>.= 
0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">ASeq</font> <a href="prob_4.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="prob_2.html#T11">PROB_2:11</a></i>
<br/>.= 
0
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:13_1_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div><b>end;</b></div><a NAME="E3:13_1"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E4:13_1"/><b>then </b><i><font color="Green">E49</font></i>: 
(  <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">ASeq</font></span>)</span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="prob_1.html#T3">PROB_1:3</a></i>;<br/><a NAME="E5:13_1"/><i><font color="Green">E50</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">ASeq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:13_1_2"/><i><font color="Green">E51</font></i>: 
 <a href="prob_3.html#K6">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font> is <a href="prob_2.html#V1">disjoint_valued</a>
 
<b>by </b><i><a class="ref" href="prob_3.html#T41">PROB_3:41</a></i>;<br/>
<a NAME="E2:13_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
 <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> <span class="p2">(<span class="default"><a href="prob_3.html#K6">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font></span>)</span></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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:13_1_2_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K6">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font></span>)</span> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ASeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="prob_3.html#T37">PROB_3:37</a></i>;<br/><b>hence </b><a NAME="E2:13_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K6">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="prob_3.html#T5">PROB_3:5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:13_1_2"/><b>then </b>
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"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="prob_3.html#K6">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>
 
<b>by </b><i><a class="ref" href="series_1.html#T17">SERIES_1:17</a></i>;<br/>
<a NAME="E5:13_1_2"/><b>then </b>
 <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> <span class="p2">(<span class="default"><a href="prob_3.html#K6">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#T32">SEQ_2:32</a></i>;<br/>
<a NAME="E6:13_1_2"/><b>then </b>
<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> <span class="p2">(<span class="default"><a href="prob_3.html#K6">@Partial_Diff_Union</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="prob_3.html#T51">PROB_3:51</a></i>;<br/>
<b>hence </b><a NAME="E7:13_1_2"/>
<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">ASeq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="prob_3.html#T40">PROB_3:40</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:13_1"/>
<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">ASeq</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:13_1_3"/>
 <a href="prob_1.html#K2">Union</a> <font color="Maroon">ASeq</font> is    <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#T58">PROB_1:58</a></i>;<br/>
<b>hence </b><a NAME="E2:13_1_3"/>
<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">ASeq</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="prob_1.html#D13">PROB_1:def 13</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:13_1"/>
<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">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:13"/><i><font color="Green">E54</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">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>reconsider </b><font color="Maroon">Y1</font> = <font color="Maroon">ASeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</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/><b>reconsider </b><font color="Maroon">Y2</font> =  <a href="prob_1.html#K2">Union</a> <font color="Maroon">ASeq</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#T58">PROB_1:58</a></i>;<br/><a NAME="E3:13_2"/>
<font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_2_1"/>
<font color="Maroon">ASeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">ASeq</font>
 
<b>by </b><i><a class="ref" href="setlim_1.html#T7">SETLIM_1:7</a></i>;<br/>
<a NAME="E2:13_2_1"/><b>then </b>
<font color="Maroon">ASeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">ASeq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E3:13_2_1"/>
<font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y2</font>
 <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="E4:13_2"/>
<font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">ASeq</font> <a href="prob_4.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>


</div>
