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

<b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Omega</font>;<br/>
<b>consider </b><font color="Maroon">P</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Sigma</font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:55_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">D</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Omega</font>  st <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> holds <br/>( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> implies <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> implies <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<a NAME="E4:55_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">A</font> being   <a href="prob_1.html#M2" target="_self">Event</a> of <font color="Maroon">Sigma</font> holds <br/> ( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> implies <font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> implies <font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> 0 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="prob_1.html#M2" target="_self">Event</a> of <font color="Maroon">Sigma</font>;<br/>

<a NAME="E1:55_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:55_1_1_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i><a class="txt" href="prob_1.html#E3:55_1_1"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:55_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">A</font> being   <a href="prob_1.html#M2" target="_self">Event</a> of <font color="Maroon">Sigma</font> holds  0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Olive">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="prob_1.html#M2" target="_self">Event</a> of <font color="Maroon">Sigma</font>;<br/>

<a NAME="E1:55_1_1_2"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="prob_1.html#E4:55_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:55_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="prob_1.html#E4:55_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:55_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">Omega</font> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:55_1_1_3"/>
<font color="Maroon">Omega</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Omega</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">O</font> = <font color="Maroon">Omega</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Omega</font> ;<br/>
<a NAME="E3:55_1_1_3"/>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:55_1_1_3"/>
<font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">Omega</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="prob_1.html#E3:55_1_1"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:55_1_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="prob_1.html#M2" target="_self">Event</a> of <font color="Maroon">Sigma</font>  st <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> holds <br/><font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="prob_1.html#K7" target="_self">\/</a> <font color="Olive">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Olive">A</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Olive">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="prob_1.html#M2" target="_self">Event</a> of <font color="Maroon">Sigma</font>;<br/><b>let </b><font color="Maroon">B</font> be    <a href="prob_1.html#M2" target="_self">Event</a> of <font color="Maroon">Sigma</font>;<br/>



<b>assume </b><a NAME="E1:55_1_1_4"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:55_1_1_4"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies ( <font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 0 ) )
 
<b>by </b><i><a class="txt" href="prob_1.html#E4:55_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E3:55_1_1_4"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="prob_1.html#K7" target="_self">\/</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:55_1_1_4"/><i><font color="Green">E94</font></i>: 
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies ( <font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 1 ) )
 
<b>by </b><i><a class="txt" href="prob_1.html#E4:55_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:55_1_1_4"/><i><font color="Green">E95</font></i>: 
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="prob_1.html#K7" target="_self">\/</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:55_1_1_4"/><i><font color="Green">E96</font></i>: 
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies ( <font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 0 ) )
 
<b>by </b><i><a class="txt" href="prob_1.html#E4:55_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E7:55_1_1_4"/>
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies  not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="prob_1.html#K7" target="_self">\/</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:55_1_1_4"/>
<font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="prob_1.html#K7" target="_self">\/</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Maroon">A</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="prob_1.html#E4:55_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="prob_1.html#D1" target="_self">Def1</a>, , , , , <a class="ref" href="prob_1.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:55_1_1"/>
for <font color="Olive">ASeq</font> being   <a href="prob_1.html#M1" target="_self">SetSequence</a> of <font color="Maroon">Sigma</font>  st <font color="Olive">ASeq</font> is <a href="prob_1.html#V2" target="_self">non-increasing</a> holds <br/>( <font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Olive">ASeq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <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" target="_self">*</a> <font color="Olive">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Olive">ASeq</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:55_1_1_5"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">ASeq</font> is <a href="prob_1.html#V2" target="_self">non-increasing</a>
 ;<br/>

<a NAME="E2:55_1_1_5"/><i><font color="Green">E98</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  <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_5_1"><b>proof </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="E1:55_1_1_5_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:55_1_1_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</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> <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:55_1_1_5"/><i><font color="Green">E99</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="hidden.html#R2">in</a> <font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font> ) implies 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" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1_1_5_2"/><i><font color="Green">E100</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="hidden.html#R2">in</a> <font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font>
 ;<br/>

<a NAME="E2:55_1_1_5_2"/>
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" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_5_2_1"><b>proof </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="E1:55_1_1_5_2_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:55_1_1_5_2_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:55_1_1_5_2_1"/><b>then </b>
<font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="prob_1.html#E3:55_1_1"><i><font color="Green">E33</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:55_1_1_5_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</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
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:55_1_1_5_2"/>
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" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:55_1_1_5"/><i><font color="Green">E102</font></i>: 
( not 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="hidden.html#R2">in</a> <font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font> implies  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_5_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1_1_5_3"/>
not 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="hidden.html#R2">in</a> <font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:55_1_1_5_3"/><i><font color="Green">E103</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Maroon">m</font>
 ;<br/>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

<a NAME="E4:55_1_1_5_3"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_5_3_1"><b>proof </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/>

<b>assume </b><a NAME="E1:55_1_1_5_3_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:55_1_1_5_3_1"/><b>then </b>
<font color="Maroon">ASeq</font> <a href="prob_1.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_1.html#K1" target="_self">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E3:55_1_1_5_3_1"/><b>then </b><i><font color="Green">E104</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:55_1_1_5_3_1"/>
<font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:55_1_1_5_3_1"/><b>then </b>
<font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="prob_1.html#E3:55_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="prob_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E6:55_1_1_5_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</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> 0
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:55_1_1_5_3"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:55_1_1_5"/><i><font color="Green">E105</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  <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1 ) implies ( <font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <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" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_5_4"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">r</font> = 1 as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>assume </b><a NAME="E2:55_1_1_5_4"/><i><font color="Green">E106</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  <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E3:55_1_1_5_4"/>
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_5_4_1"><b>proof </b></a><div class="add">

<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E1:55_1_1_5_4_1"/>
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"><font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:55_1_1_5_4"/>
( <font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <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" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:55_1_1_5"/><i><font color="Green">E107</font></i>: 
(  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 0 implies ( <font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <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" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> 0 ) )
 
<b>by </b><i/>;<br/>
<a NAME="E7:55_1_1_5"/><i><font color="Green">E108</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="hidden.html#R2">in</a> <font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font> ) implies ( <font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <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" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">ASeq</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_5_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1_1_5_5"/><i><font color="Green">E109</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="hidden.html#R2">in</a> <font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font>
 ;<br/>

<a NAME="E2:55_1_1_5_5"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">ASeq</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:55_1_1_5_5"/>
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">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:55_1_1_5_5"/><b>then </b>
 <a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">ASeq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:55_1_1_5_5"/>
( <font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <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" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">ASeq</font></span>)</span> )
 <b>by </b><i><a class="txt" href="prob_1.html#E3:55_1_1"><i><font color="Green">E33</font></i></a>, , , <a class="ref" href="prob_1.html#T6" target="_self">Th6</a>, <a class="ref" href="prob_1.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:55_1_1_5"/>
( not 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="hidden.html#R2">in</a> <font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font> implies ( <font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <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" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">ASeq</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_5_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1_1_5_6"/><i><font color="Green">E111</font></i>: 
not 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="hidden.html#R2">in</a> <font color="Maroon">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font>
 ;<br/>

<a NAME="E2:55_1_1_5_6"/><b>then </b><i><font color="Green">E112</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">ASeq</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:55_1_1_5_6"/>
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">ASeq</font> <a href="prob_1.html#K1" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:55_1_1_5_6"/><b>then </b>
 <a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">ASeq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:55_1_1_5_6"/>
( <font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <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" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">ASeq</font></span>)</span> )
 <b>by </b><i><a class="txt" href="prob_1.html#E3:55_1_1"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="prob_1.html#T4" target="_self">Th4</a>, <a class="ref" href="prob_1.html#T6" target="_self">Th6</a>, <a class="ref" href="prob_1.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:55_1_1_5"/>
( <font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Maroon">ASeq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <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" target="_self">*</a> <font color="Maroon">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Maroon">ASeq</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:55_1_1"/>
( ( for <font color="Olive">A</font> being   <a href="prob_1.html#M2" target="_self">Event</a> of <font color="Maroon">Sigma</font> holds  0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Olive">A</font> ) &amp; <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">Omega</font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="prob_1.html#M2" target="_self">Event</a> of <font color="Maroon">Sigma</font>  st <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> holds <br/><font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="prob_1.html#K7" target="_self">\/</a> <font color="Olive">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Olive">A</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10" target="_self">.</a> <font color="Olive">B</font></span>)</span> ) &amp; ( for <font color="Olive">ASeq</font> being   <a href="prob_1.html#M1" target="_self">SetSequence</a> of <font color="Maroon">Sigma</font>  st <font color="Olive">ASeq</font> is <a href="prob_1.html#V2" target="_self">non-increasing</a> holds <br/>( <font color="Maroon">P</font> <a href="prob_1.html#K9" target="_self">*</a> <font color="Olive">ASeq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <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" target="_self">*</a> <font color="Olive">ASeq</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4" target="_self">Intersection</a> <font color="Olive">ASeq</font></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="prob_1.html#E5:55_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="prob_1.html#E6:55_1_1"><i><font color="Green">E36</font></i></a>, </i>;<br/>


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