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<b>let </b><font color="Maroon" title="c1">E</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">C</font> being  <a href="rpr_1.html#NM2" title="RPR_1:NM.2">Event</a> of <font color="Maroon" title="c1">E</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">B</font> &amp; <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b2">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">C</font> holds <br/> <a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Olive" title="b2">B</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Olive" title="b3">C</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</font> be   <a href="rpr_1.html#NM2" title="RPR_1:NM.2">Event</a> of <font color="Maroon" title="c1">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c3">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">C</font> implies  <a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c4">C</font></span>)</span> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:35"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font>
 <b>and </b><br/><a NAME="E2:35"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">C</font>
 <b>and </b><br/><a NAME="E3:35"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c4">C</font></span>)</span></span><br/>

<a NAME="E4:35"/><i><font color="Green" title="E21">A4</font></i>: 
 <a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="rpr_1.html#T41" target="_self" title="RPR_1:th.41">Th41</a>, <a class="ref" href="xboole_1.html#T74" title="XBOOLE_1:th.74">XBOOLE_1:74</a></i>;<br/>
 <a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rpr_1.html#T56" target="_self" title="RPR_1:th.56">Th56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E4:35"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E3:35"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="rpr_1.html#T41" target="_self" title="RPR_1:th.41">Th41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E2:35"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="rpr_1.html#T41" target="_self" title="RPR_1:th.41">Th41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E1:35"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="rpr_1.html#T41" target="_self" title="RPR_1:th.41">Th41</a></i>
;<br/>
<b>hence </b><a NAME="E6:35"/>
 <a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rpr_1.html#K1" title="RPR_1:func.1">prob</a> <font color="Maroon" title="c4">C</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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