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<b>let </b><font color="Maroon">Omega</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Sigma</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="prob_1.html#M3">Probability</a> of <font color="Maroon">Sigma</font>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>thus </b><a NAME="E1:51"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="prob_4.html#K7" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</font> implies  ex <font color="Olive">A1</font>, <font color="Olive">A2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Olive">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Olive">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A2</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">A2</font> <a href="subset_1.html#K6">\</a> <font color="Olive">A1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="prob_4.html#K7" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:51_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp;  ex <font color="Olive">C</font> being   <a href="prob_4.html#M2" target="_self">thin</a> of <font color="Maroon">P</font> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">C</font> being    <a href="prob_4.html#M2" target="_self">thin</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E5:51_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">D</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:51_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="prob_4.html#E3:51_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">Sigma</font> <b>by </b><i>, <a class="ref" href="prob_1.html#D10">PROB_1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">Sigma</font> <b>by </b><i>, <a class="ref" href="prob_1.html#D10">PROB_1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">D</font> <a href="prob_1.html#K7">\/</a> <font color="Maroon">B</font> as    <a href="prob_1.html#M2">Event</a> of <font color="Maroon">Sigma</font> ;<br/>
<a NAME="E11:51_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#D10">PROB_1:def 10</a></i>;<br/>
<a NAME="E12:51_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E7:51_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E13:51_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="txt" href="prob_4.html#E7:51_1"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>
<i><font color="Green">E51</font></i>: <font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span> = 
<font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="prob_1.html#K7">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E15:51_1"/>
<font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="prob_1.html#K7">\/</a> <font color="Maroon">B</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <font color="Maroon">D</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">.</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_1.html#T75">PROB_1:75</a></i>;<br/>
<a NAME="E16:51_1"/><b>then </b>
<font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="prob_4.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T49">XREAL_1:49</a></i>;<br/>
<a NAME="E17:51_1"/><b>then </b>
<font color="Maroon">P</font> <a href="prob_1.html#K10">.</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="prob_1.html#K8">\</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="prob_1.html#D13">PROB_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E18:51_1"/>
 ex <font color="Olive">A1</font>, <font color="Olive">A2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Olive">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Olive">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A2</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">A2</font> <a href="subset_1.html#K6">\</a> <font color="Olive">A1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , , </i>;<br/>


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

<b>thus </b><a NAME="E2:51"/>
(  ex <font color="Olive">A1</font>, <font color="Olive">A2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Olive">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Olive">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A2</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">A2</font> <a href="subset_1.html#K6">\</a> <font color="Olive">A1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 ) implies <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="prob_4.html#K7" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">A1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">A2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:51_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Maroon">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sigma</font> &amp; <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A1</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Sigma</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">A2</font> = <font color="Maroon">A2</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Sigma</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">C</font> = <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A1</font>;<br/>
<a NAME="E4:51_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E5:51_2"/>
<font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A1</font> is    <a href="prob_4.html#M2" target="_self">thin</a> of <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">A2</font> <a href="measure1.html#K4">\</a> <font color="Maroon">A1</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Sigma</font> ;<br/>
<a NAME="E2:51_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:51_2_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A1</font>
 ;<br/>

<a NAME="E2:51_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:51_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:51_2_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Omega</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E4:51_2_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A1</font> is    <a href="prob_4.html#M2" target="_self">thin</a> of <font color="Maroon">P</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:51_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="prob_4.html#K7" target="_self">COM</a> <font color="Maroon">Sigma</font>,<font color="Maroon">P</font>
 <b>by </b><i>, </i>;<br/>


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


</div>
