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<span class="kw">let </span><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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Sigma</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">Omega</font><br/>  for <font color="Olive" title="b2">P</font> being    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Olive" title="b1">Sigma</font> holds   <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Olive" title="b1">Sigma</font>,<font color="Olive" title="b2">P</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Olive" title="b1">Sigma</font>,<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Olive" title="b2">P</font></span>)</span>)</span><br/><span class="kw">let </span><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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Maroon" title="c2">Sigma</font> holds   <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Olive" title="b1">P</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Olive" title="b1">P</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be    <a href="prob_1.html#M2" title="PROB_1:mode.2">Probability</a> of <font color="Maroon" title="c2">Sigma</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>)</span><br/>





<a NAME="E1:33"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>) or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">B</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:33_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 <span class="kw">and </span><br/><a NAME="E4:33_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 ex <font color="Olive" title="b1">C</font> being    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure3.html#D3" onmouseover="rs('measure3/D3')" onmouseout="rh()">MEASURE3:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">C</font> being    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font><span class="kw"> such that </span><br/><a NAME="E6:33_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">C1</font> = <font color="Maroon" title="c6">C</font> as    <a href="prob_4.html#M1" title="PROB_4:mode.1">thin</a> of <font color="Maroon" title="c3">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#T23" target="_self" onmouseover="rs('prob_4/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<a NAME="E8:33_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:33_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="prob_4.html#D5" target="_self" onmouseover="rs('prob_4/D5')" onmouseout="rh()">Def5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:33"/>
 <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="33_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>) or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">B</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:33_2"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 <span class="kw">and </span><br/><a NAME="E4:33_2"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
 ex <font color="Olive" title="b1">C</font> being    <a href="prob_4.html#M1" title="PROB_4:mode.1">thin</a> of <font color="Maroon" title="c3">P</font> st <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#D5" target="_self" onmouseover="rs('prob_4/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">C</font> being    <a href="prob_4.html#M1" title="PROB_4:mode.1">thin</a> of <font color="Maroon" title="c3">P</font><span class="kw"> such that </span><br/><a NAME="E6:33_2"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_2"><span class="lab"><font color="Green" title="E29">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">C1</font> = <font color="Maroon" title="c6">C</font> as    <a href="measure3.html#M1" title="MEASURE3:mode.1">thin</a> of  <a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_4.html#T23" target="_self" onmouseover="rs('prob_4/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<a NAME="E8:33_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_2"><span class="lab"><font color="Green" title="E30">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:33_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_2"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="measure3.html#D3" onmouseover="rs('measure3/D3')" onmouseout="rh()">MEASURE3:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:33"/>
 <a href="prob_4.html#K4" title="PROB_4:func.4">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<font color="Maroon" title="c3">P</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure3.html#K2" title="MEASURE3:func.2">COM</a> (<font color="Maroon" title="c2">Sigma</font>,<span class="p1">(<span class="default"><a href="prob_4.html#K2" title="PROB_4:func.2">P2M</a> <font color="Maroon" title="c3">P</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E27">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
