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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="rpr_1.html#NM2">Event</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green">E25</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="rpr_1.html#K1" target="_self">prob</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:47"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="subset_1.html#T25">SUBSET_1:25</a></i>;<br/>
<a NAME="E3:47"/>
<span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<a NAME="E1:47_1"/>
<span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E2:47_1"/>
<span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:47"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rpr_1.html#E2:47"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E5:47"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="rpr_1.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E6:47"/><b>then </b>
<span class="p1">(<span class="default"><a href="rpr_1.html#K1" target="_self">prob</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rpr_1.html#K1" target="_self">prob</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rpr_1.html#K1" target="_self">prob</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rpr_1.html#E4:47"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rpr_1.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E7:47"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rpr_1.html#K2" target="_self">prob</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rpr_1.html#K1" target="_self">prob</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rpr_1.html#K1" target="_self">prob</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rpr_1.html#K1" target="_self">prob</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rpr_1.html#E1:47"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rpr_1.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E8:47"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rpr_1.html#K2" target="_self">prob</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rpr_1.html#K1" target="_self">prob</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rpr_1.html#K2" target="_self">prob</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rpr_1.html#K1" target="_self">prob</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rpr_1.html#K1" target="_self">prob</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rpr_1.html#E1:47"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rpr_1.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E9:47"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rpr_1.html#K2" target="_self">prob</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rpr_1.html#K2" target="_self">prob</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rpr_1.html#K1" target="_self">prob</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rpr_1.html#K1" target="_self">prob</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E10:47"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rpr_1.html#K2" target="_self">prob</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="rpr_1.html#K2" target="_self">prob</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rpr_1.html#K1" target="_self">prob</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rpr_1.html#K1" target="_self">prob</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="rpr_1.html#E1:47"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/>
<a NAME="E11:47"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rpr_1.html#K2" target="_self">prob</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rpr_1.html#K2" target="_self">prob</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rpr_1.html#K1" target="_self">prob</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rpr_1.html#K1" target="_self">prob</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
;<br/>
<a NAME="E12:47"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rpr_1.html#K2" target="_self">prob</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="rpr_1.html#K2" target="_self">prob</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="rpr_1.html#E1:47"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E13:47"/>
(  <a href="rpr_1.html#K2" target="_self">prob</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rpr_1.html#K2" target="_self">prob</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="rpr_1.html#K2" target="_self">prob</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rpr_1.html#K2" target="_self">prob</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> )
 ;<br/>


</div>
