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<b>let </b><font color="Maroon" title="c1">Omega</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Sigma</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font>;<br/><b>let </b><font color="Maroon" title="c3">P</font> be    <a href="prob_1.html#M3" title="PROB_1:mode.3">Probability</a> of <font color="Maroon" title="c2">Sigma</font>;<br/><b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</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><b>that </b><br/><a NAME="E1:38"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font>
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c4">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>and </b><br/><a NAME="E3:38"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="prob_1.html#K10" title="PROB_1:func.10">.</a> <font color="Maroon" title="c5">B</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/>

<a NAME="E4:38"/><i><font color="Green" title="E28">A4</font></i>: 
<span class="p1">(<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#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c4">A</font>,<span class="p1">(<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#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c5">B</font> <a href="prob_2.html#R1" title="PROB_2:pred.1">are_independent_respect_to</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="prob_2.html#E1:38"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="prob_2.html#T38" target="_self" title="PROB_2:th.38">Th38</a></i>;<br/>
<a NAME="E5:38"/><i><font color="Green" title="E29">A5</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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"><span class="p2">(<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#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c4">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_2.html#E2:38"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="prob_2.html#T27" target="_self" title="PROB_2:th.27">Th27</a></i>;<br/>
<a NAME="E6:38"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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"><span class="p2">(<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#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c5">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_2.html#E3:38"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="prob_2.html#T27" target="_self" title="PROB_2:th.27">Th27</a></i>;<br/>
<a NAME="E7:38"/><b>then </b><i><font color="Green" title="E30">A6</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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"><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#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c4">A</font></span>)</span></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"><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#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c5">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="prob_2.html#E5:38"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/>
<div><i><font color="Green" title="E31">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">r</font>, <font color="Maroon" title="c7">r1</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:38_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r1</font>
 ;<br/><a NAME="E2:38_1"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><a NAME="E3:38_1"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c7">r1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E4:38_1"/>
<font color="Maroon" title="c6">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font>
 ;<br/></div><b>end;</b></div>
<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="c4">A</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <font color="Maroon" title="c5">B</font></span>)</span> = 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</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"><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#K8" title="PROB_1:func.8">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <font color="Maroon" title="c5">B</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prob_2.html#T26" target="_self" title="PROB_2:th.26">Th26</a></i>
<br/>.= 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</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"><span class="p3">(<span class="default"><span class="p4">(<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#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="prob_1.html#K6" title="PROB_1:func.6">/\</a> <span class="p3">(<span class="default"><span class="p4">(<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#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c5">B</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T53" title="XBOOLE_1:th.53">XBOOLE_1:53</a></i>
<br/>.= 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</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"><span class="p4">(<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#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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"><span class="p4">(<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#K8" title="PROB_1:func.8">\</a> <font color="Maroon" title="c5">B</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prob_2.html#E4:38"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="prob_2.html#D5" target="_self" title="PROB_2:def.5">Def5</a></i>
;<br/>
<b>hence </b><a NAME="E10:38"/>
<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="c4">A</font> <a href="prob_1.html#K7" title="PROB_1:func.7">\/</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="prob_2.html#E7:38"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="prob_2.html#E8:38"><i><font color="Green" title="E31">A7</font></i></a></i>;<br/>


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