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







<b>assume </b><a NAME="E1:18"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">ASeq</font> is <a href="prob_1.html#V3">non-decreasing</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">V</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/>
<b>set </b><font color="Maroon">BSeq</font> =  <a href="prob_2.html#K3" target="_self">@Complement</a> <font color="Maroon">ASeq</font>;<br/>
<a NAME="E3:18"/><i><font color="Green">E38</font></i>: 
 <a href="prob_2.html#K3" target="_self">@Complement</a> <font color="Maroon">ASeq</font> is <a href="prob_1.html#V2">non-increasing</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:18"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="prob_2.html#K3" target="_self">@Complement</a> <font color="Maroon">ASeq</font></span>)</span> is <a href="seq_2.html#V4">convergent</a>
 
<b>by </b><i><a class="ref" href="prob_1.html#D13">PROB_1:def 13</a></i>;<br/>
<a NAME="E5:18"/>
 <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="prob_2.html#K3" target="_self">@Complement</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">ASeq</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4">Intersection</a> <span class="p2">(<span class="default"><a href="prob_2.html#K3" target="_self">@Complement</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_1.html#T68">PROB_1:68</a></i>;<br/>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_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/><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_2.html#K3" target="_self">@Complement</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_2.html#K3" target="_self">@Complement</a> <font color="Maroon">ASeq</font></span>)</span> <a href="prob_2.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ASeq</font> <a href="prob_2.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="prob_1.html#D4">PROB_1:def 4</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">Sigma</font></span>)</span> <a href="prob_1.html#K8">\</a> <span class="p2">(<span class="default"><font color="Maroon">ASeq</font> <a href="prob_2.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>
<br/>.= 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">ASeq</font> <a href="prob_2.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prob_1.html#T68">PROB_1:68</a></i>
<br/>.= 
1 <a href="real_1.html#K5">-</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="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><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> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E2:18_1"/>
<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="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</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> <span class="p3">(<span class="default"><a href="prob_2.html#K3" target="_self">@Complement</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:18"/>
<font color="Maroon">P</font> <a href="prob_1.html#K9">*</a> <font color="Maroon">ASeq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i>, </i>;<br/>

<b>thus </b> <a href="seq_2.html#K2">lim</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> = 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</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_2.html#K3" target="_self">@Complement</a> <font color="Maroon">ASeq</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">V</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="prob_1.html#D13">PROB_1:def 13</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <font color="Maroon">ASeq</font></span>)</span>

;<br/>


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