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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">A</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font> ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span> );<br/>
<a NAME="E1:39_1_1"/><i><font color="Green" title="E26">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font> <b>by </b><i><a class="ref" href="prob_1.html#T48" title="PROB_1:th.48">PROB_1:48</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:39_1_1_1"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
 ;<br/>
<b>take </b>
<font color="Maroon" title="c7">y</font>
;<br/>

<b>thus </b><a NAME="E5:39_1_1_1"/>
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">y</font>] )
 <b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1_1"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">Sigma</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green" title="E27">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="prob_2.html#E1:39_1_1"><i><font color="Green" title="E26">A2</font></i></a>);<br/></i>
<a NAME="E4:39_1_1"/><i><font color="Green" title="E28">A5</font></i>: 
for <font color="Olive" title="b1">A</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font> holds  <font color="Maroon" title="c5">f</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</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" title="c6">A</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font>;<br/>

<a NAME="E1:39_1_1_2"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D10" title="PROB_1:def.10">PROB_1:def 10</a></i>;<br/>
<a NAME="E2:39_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">C</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font> ex <font color="Olive" title="b2">r1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Olive" title="b2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Olive" title="b2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">C</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="prob_2.html#E3:39_1_1"><i><font color="Green" title="E27">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:39_1_1"/>
<font color="Maroon" title="c5">f</font> is    <a href="prob_1.html#M3" title="PROB_1:mode.3">Probability</a> of <font color="Maroon" title="c2">Sigma</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" title="b1">A</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font> holds  0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Olive" title="b1">A</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="prob_1.html#D13" title="PROB_1:def.13">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" title="c6">A</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font>;<br/>

<a NAME="E1:39_1_1_3_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_1.html#D13" title="PROB_1:def.13">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" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_2.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="xreal_1.html#T138" title="XREAL_1:th.138">XREAL_1:138</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1_3_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/>


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

<b>thus </b><font color="Maroon" title="c5">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">Omega</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_1.html#K5" title="PROB_1:func.5">[#]</a> <font color="Maroon" title="c2">Sigma</font></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green" title="E28">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E6"><i><font color="Green" title="E3">Lm1</font></i></a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="prob_2.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
;<br/>

<b>thus </b><a NAME="E3:39_1_1_3"/>
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">C</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font>  st <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">C</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <font color="Olive" title="b2">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Olive" title="b2">C</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" title="c6">A</font>, <font color="Maroon" title="c7">C</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font>;<br/>



<b>assume </b><a NAME="E1:39_1_1_3_3"/>
<font color="Maroon" title="c6">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">C</font>
 ;<br/>

<a NAME="E2:39_1_1_3_3"/><b>then </b><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">C</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T76" title="XBOOLE_1:th.76">XBOOLE_1:76</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c5">f</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <font color="Maroon" title="c7">C</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <font color="Maroon" title="c7">C</font></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green" title="E28">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E2:39_1_1_3_3"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="prob_1.html#D13" title="PROB_1:def.13">PROB_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63" title="XCMPLX_1:th.63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c7">C</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green" title="E28">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c7">C</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green" title="E28">A5</font></i></a></i>
;<br/>


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

<b>thus </b><a NAME="E4:39_1_1_3"/>
for <font color="Olive" title="b1">ASeq</font> being   <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>  st <font color="Olive" title="b1">ASeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-increasing</a> holds <br/>( <font color="Maroon" title="c5">f</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Olive" title="b1">ASeq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Olive" title="b1">ASeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Olive" title="b1">ASeq</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" title="c6">ASeq</font> be    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font>;<br/>

<b>assume </b><a NAME="E1:39_1_1_3_4"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c6">ASeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-increasing</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">BSeq</font> being    <a href="prob_1.html#M1" title="PROB_1:mode.1">SetSequence</a> of <font color="Maroon" title="c2">Sigma</font><b> such that </b><br/><a NAME="E3:39_1_1_3_4"/><i><font color="Green" title="E30">A8</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c7">BSeq</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">ASeq</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="prob_2.html#T9" target="_self" title="PROB_2:th.9">Th9</a></i>;<br/>
<a NAME="E4:39_1_1_3_4"/><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c7">BSeq</font> is <a href="prob_1.html#V2" title="PROB_1:attr.2">non-increasing</a>
 
<b>by </b><i><a class="txt" href="prob_2.html#E1:39_1_1_3_4"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="prob_2.html#E3:39_1_1_3_4"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="prob_2.html#T10" target="_self" title="PROB_2:th.10">Th10</a></i>;<br/>
<a NAME="E5:39_1_1_3_4"/><i><font color="Green" title="E32">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:39_1_1_3_4"/><i><font color="Green" title="E33">A11</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">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" title="c8">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1_3_4_1"/><i><font color="Green" title="E34">A12</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">n1</font> = <font color="Maroon" title="c8">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="prob_2.html#E5:39_1_1_3_4"><i><font color="Green" title="E32">A10</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> = 
<font color="Maroon" title="c5">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">ASeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</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" title="E34">A12</font></i></a>, <a class="txt" href="prob_2.html#E5:39_1_1_3_4"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="prob_2.html#T11" target="_self" title="PROB_2:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">ASeq</font> <a href="prob_2.html#K1" title="PROB_2:func.1">.</a> <font color="Maroon" title="c9">n1</font></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green" title="E28">A5</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">BSeq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E3:39_1_1_3_4"><i><font color="Green" title="E30">A8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</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" title="E34">A12</font></i></a>, <a class="txt" href="prob_2.html#E5:39_1_1_3_4"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="prob_2.html#T11" target="_self" title="PROB_2:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">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" title="E34">A13</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="seq_1.html#K14" title="SEQ_1:func.14">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_2.html#E5:39_1_1_3_4"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="prob_2.html#E6:39_1_1_3_4"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E9:39_1_1_3_4"/><i><font color="Green" title="E35">A14</font></i>: 
( <font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c7">BSeq</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1_3_4"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="prob_1.html#D13" title="PROB_1:def.13">PROB_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E10:39_1_1_3_4"/>
<font color="Maroon" title="c5">f</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="prob_2.html#E8:39_1_1_3_4"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="seq_2.html#T21" title="SEQ_2:th.21">SEQ_2:21</a></i>;<br/>

<b>thus </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="prob_1.html#K9" title="PROB_1:func.9">*</a> <font color="Maroon" title="c6">ASeq</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p2">(<span class="default"><a href="prob_2.html#K2" title="PROB_2:func.2">@Intersection</a> <font color="Maroon" title="c7">BSeq</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" title="E34">A13</font></i></a>, <a class="txt" href="prob_2.html#E9:39_1_1_3_4"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="seq_2.html#T22" title="SEQ_2:th.22">SEQ_2:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p2">(<span class="default"><a href="prob_2.html#K2" title="PROB_2:func.2">@Intersection</a> <font color="Maroon" title="c7">BSeq</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prob_2.html#K2" title="PROB_2:func.2">@Intersection</a> <font color="Maroon" title="c6">ASeq</font></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E3:39_1_1_3_4"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="prob_2.html#T12" target="_self" title="PROB_2:th.12">Th12</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c6">ASeq</font></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green" title="E28">A5</font></i></a></i>
;<br/>


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


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">f</font> as    <a href="prob_1.html#M3" title="PROB_1:mode.3">Probability</a> of <font color="Maroon" title="c2">Sigma</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">f</font>
;<br/>

<b>thus </b><a NAME="E7:39_1_1"/>
for <font color="Olive" title="b1">A</font> being   <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c2">Sigma</font> holds  <font color="Maroon" title="c6">f</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">A</font> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <font color="Maroon" title="c4">B</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="prob_2.html#E4:39_1_1"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/>


</div>
