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<div class="add">

<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#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" 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">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Omega</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b2">f</font> <a href="finance1.html#R1" title="FINANCE1:pred.1">is_random_variable_on</a> <font color="Olive" title="b1">Sigma</font>, <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  holds <br/>( <font color="Olive" title="b2">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">[#]</a> <font color="Olive" title="b1">Sigma</font> &amp; <font color="Olive" title="b2">f</font> is    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Olive" title="b1">Sigma</font> )</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Omega</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b1">f</font> <a href="finance1.html#R1" title="FINANCE1:pred.1">is_random_variable_on</a> <font color="Maroon" title="c2">Sigma</font>, <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  holds <br/>( <font color="Olive" title="b1">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">[#]</a> <font color="Maroon" title="c2">Sigma</font> &amp; <font color="Olive" title="b1">f</font> is    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">Omega</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="finance1.html#R1" title="FINANCE1:pred.1">is_random_variable_on</a> <font color="Maroon" title="c2">Sigma</font>, <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  implies ( <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">[#]</a> <font color="Maroon" title="c2">Sigma</font> &amp; <font color="Maroon" title="c3">f</font> is    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:31"/><span class="lab"><font color="Green" title="E19">A0</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="finance1.html#R1" title="FINANCE1:pred.1">is_random_variable_on</a> <font color="Maroon" title="c2">Sigma</font>, <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">[#]</a> <font color="Maroon" title="c2">Sigma</font> &amp; <font color="Maroon" title="c3">f</font> is    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font> )</span><br/>

<a NAME="E2:31"/><span class="lab"><font color="Green" title="E20">J0</font></span>: 
 for <font color="Olive" title="b1">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <font color="Maroon" title="c1">Omega</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">Omega</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font></span><br/>

<a NAME="E1:31_1"/>
 <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Omega</font> : <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E19">A0</font></span></a>, <a class="ref" href="finance1.html#T9" target="_self" title="FINANCE1:th.9">Th4</a></span>;<br/>
<a NAME="E2:31_1"/><span class="kw">then </span>
 <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">r</font>) is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Sigma</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E19">A0</font></span></a>, <a class="ref" href="finance1.html#T9" target="_self" title="FINANCE1:th.9">Th4</a></span>;<br/>
<a NAME="E3:31_1"/><span class="kw">then </span>
 <a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">r</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E4:31_1"/>
<font color="Maroon" title="c1">Omega</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>; <a class="txt" 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="E3:31"/><span class="lab"><font color="Green" title="E21">Fin0</font></span>: 
 for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <span class="p1">(<span class="default"><a href="prob_1.html#K4" title="PROB_1:func.4">[#]</a> <font color="Maroon" title="c2">Sigma</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="prob_1.html#K4" title="PROB_1:func.4">[#]</a> <font color="Maroon" title="c2">Sigma</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">r</font> = <font color="Maroon" title="c4">r</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E2:31_2"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K4" title="PROB_1:func.4">[#]</a> <font color="Maroon" title="c2">Sigma</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E20">J0</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:31_2"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K4" title="PROB_1:func.4">[#]</a> <font color="Maroon" title="c2">Sigma</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K12" title="MESFUNC1:func.12">less_dom</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">r</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Sigma</font>
 ; <a class="txt" 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="E4:31"/>
<font color="Maroon" title="c3">f</font> is    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:31_3"/>
 ex <font color="Olive" title="b1">X</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Sigma</font> st <br/>( <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font> &amp; <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c4">X</font> =  <a href="prob_1.html#K4" title="PROB_1:func.4">[#]</a> <font color="Maroon" title="c2">Sigma</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Sigma</font> ;<br/>
<a NAME="E2:31_3_1"/>
<font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E21">Fin0</font></span></a>, <a class="ref" href="mesfunc6.html#T12" title="MESFUNC6:th.12">MESFUNC6:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:31_3_1"/>
 ex <font color="Olive" title="b1">X</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">Sigma</font> st <br/>( <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Omega</font> &amp; <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Olive" title="b1">X</font> )
 ; <a class="txt" 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="E2:31_3"/>
<font color="Maroon" title="c3">f</font> is    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="random_1.html#D2" title="RANDOM_1:def.2">RANDOM_1:def 2</a></span>; <a class="txt" 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="E5:31"/>
( <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a>  <a href="prob_1.html#K4" title="PROB_1:func.4">[#]</a> <font color="Maroon" title="c2">Sigma</font> &amp; <font color="Maroon" title="c3">f</font> is    <a href="random_1.html#M1" title="RANDOM_1:mode.1">Real-Valued-Random-Variable</a> of <font color="Maroon" title="c2">Sigma</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E21">Fin0</font></span></a>, <a class="ref" href="mesfunc6.html#T12" title="MESFUNC6:th.12">MESFUNC6:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
