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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> );<br/>
<a NAME="E1:39_1_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="prob_1.html#T48">PROB_1:48</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:39_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E5:39_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>] )
 <b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="prob_2.html#E1:39_1_1"><i><font color="Green">E26</font></i></a>);<br/></i>
<a NAME="E4:39_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K10">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E1:39_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D10">PROB_1:def 10</a></i>;<br/>
<a NAME="E2:39_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="prob_2.html#E3:39_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:39_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:39_1_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font> holds 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K10">.</a> <font color="Olive">b<sub>1</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="prob_1.html#D13">PROB_1:def 13</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E1:39_1_1_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_1.html#D13">PROB_1:def 13</a></i>;<br/>
<a NAME="E2:39_1_1_3_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_2.html#E1:39"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xreal_1.html#T138">XREAL_1:138</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1_3_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_1.html#K5">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="prob_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="prob_2.html#E1:39"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/>

<b>thus </b><a NAME="E3:39_1_1_3"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="prob_1.html#K7">\/</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K10">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K10">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E1:39_1_1_3_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:39_1_1_3_3"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>7</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T76">XBOOLE_1:76</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="prob_1.html#K7">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="prob_1.html#K7">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K7">\/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E2:39_1_1_3_3"><i><font color="Green">E29</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>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green">E28</font></i></a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:39_1_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="prob_1.html#V2">non-increasing</a> holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4">Intersection</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:39_1_1_3_4"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="prob_1.html#V2">non-increasing</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:39_1_1_3_4"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>7</sub></font> <a href="prob_2.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="prob_2.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="prob_2.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E4:39_1_1_3_4"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="prob_1.html#V2">non-increasing</a>
 
<b>by </b><i><a class="txt" href="prob_2.html#E1:39_1_1_3_4"><i><font color="Green">E29</font></i></a>, <a class="txt" href="prob_2.html#E3:39_1_1_3_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="prob_2.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E5:39_1_1_3_4"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K1">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>6</sub></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/>
<a NAME="E6:39_1_1_3_4"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>7</sub></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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1_3_4_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">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>6</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="prob_2.html#E5:39_1_1_3_4"><i><font color="Green">E32</font></i></a></i>;<br/><b>thus </b><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>6</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E1:39_1_1_3_4_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="prob_2.html#E5:39_1_1_3_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="prob_2.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="prob_2.html#K1" target="_self">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E3:39_1_1_3_4"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E1:39_1_1_3_4_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="prob_2.html#E5:39_1_1_3_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="prob_2.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E8:39_1_1_3_4"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="seq_1.html#K14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_2.html#E5:39_1_1_3_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="prob_2.html#E6:39_1_1_3_4"><i><font color="Green">E33</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E9:39_1_1_3_4"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>7</sub></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">c<sub>3</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4">Intersection</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1_3_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="prob_1.html#D13">PROB_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E10:39_1_1_3_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="prob_1.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font> is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="txt" href="prob_2.html#E8:39_1_1_3_4"><i><font color="Green">E34</font></i></a>, <a class="ref" href="seq_2.html#T21">SEQ_2:21</a></i>;<br/>

<b>thus </b> <a href="seq_2.html#K2">lim</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>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><a href="prob_2.html#K2" target="_self">@Intersection</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E8:39_1_1_3_4"><i><font color="Green">E34</font></i></a>, <a class="txt" href="prob_2.html#E9:39_1_1_3_4"><i><font color="Green">E35</font></i></a>, <a class="ref" href="seq_2.html#T22">SEQ_2:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><a href="prob_2.html#K2" target="_self">@Intersection</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_2.html#K2" target="_self">@Intersection</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E3:39_1_1_3_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="prob_2.html#T12" target="_self">Th12</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4">Intersection</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green">E28</font></i></a></i>
;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E7:39_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="prob_1.html#M2">Event</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>6</sub></font> <a href="prob_1.html#K10">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="prob_1.html#K6">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="prob_1.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div>
