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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>









<b>assume </b><a NAME="E1:39"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="prob_2.html#V1">disjoint_valued</a>
 ;<br/>

<a NAME="E2:39"/><i><font color="Green">E36</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" target="_self">@Partial_Union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="seq_1.html#T3">SEQ_1:3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" target="_self">@Partial_Union</a> <font color="Maroon">c<sub>4</sub></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> <span class="p1">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E3:39"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="prob_3.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E4:39"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" target="_self">@Partial_Union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</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">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:39_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:39_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="prob_3.html#K5" target="_self">@Partial_Union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>4</sub></font> <a href="kurato_2.html#K7">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prob_3.html#E1:39"><i><font color="Green">E35</font></i></a>, <a class="ref" href="prob_3.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E4:39_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="kurato_2.html#K7">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="prob_3.html#K5" target="_self">@Partial_Union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="prob_1.html#T57">PROB_1:57</a></i>;<br/>
<i><font color="Green">E42</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" target="_self">@Partial_Union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K5" target="_self">@Partial_Union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="kurato_2.html#K7">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_3.html#E2:39"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K5" target="_self">@Partial_Union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="kurato_2.html#K7">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prob_3.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_3.html#K5" target="_self">@Partial_Union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="kurato_2.html#K7">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="kurato_2.html#K7">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_3.html#E4:39_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="prob_3.html#E3:39_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="prob_1.html#D13">PROB_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="prob_3.html#K5" target="_self">@Partial_Union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="kurato_2.html#K7">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_3.html#E2:39"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<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">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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/>.= 
<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">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="kurato_2.html#K7">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_3.html#E2:39"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/>
<b>hence </b><a NAME="E7:39_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="prob_3.html#E5:39_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="prob_3.html#E1:39_1"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:39"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="prob_3.html#E3:39"><i><font color="Green">E37</font></i></a>, <a class="txt" href="prob_3.html#E4:39"><i><font color="Green">E38</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:39"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="prob_3.html#K5" target="_self">@Partial_Union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</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">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
